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PROJECTIVE DIFFERENTIAL GEOMETRY 
OF CURVES AND SURFACES 



THE UNIVERSITY OF CHICAGO PRESS 
CHICAGO, ILLINOIS 



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NKW YORK 

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THE COMMERCIAL PRESS, LIMITED 

SHANGHAI 



PROJECTIVE DIFFERENTIAL 

GEOMETRY OF CURVES 

AND SURFACES 



By 
ERNEST PRESTON LANE 

Professor of Mathematics in the 
University of Chicago 




THE UNIVERSITY OF CHICAGO PRESS 
CHICAGO ILLINOIS 



COPYRIGHT 1932 BY THE UNIVERSITY OF CHICAGO 
ALL RIGHTS RESERVED. PUBLISHED JUNE 1932 



) AND PRINTED BY THE UNIVERSITY OF CHICAGO PRE 
CHICAGO, ILLINOIS, U.S.A. 



PREFACE 

Projective Differential Geometry is largely a product of the first three 
decades of the twentieth century. The theory has been developed in five 
or more different languages, by three or four well-recognized methods, in 
various and sundry notations, and has been published partly in journals 
not readily accessible to all. 

The author's aim in writing this book was to organize an exposition of 
these researches. He desired to coordinate the "results achieved on both 
sides of the Atlantic so as to make the work of the European geometers 
more readily accessible to American students, and so as to make better 
known to others the accomplishments of the American school. The author 
has made use of those of his own results which have been published in jour- 
nals, and has also included occasional new results hitherto unpublished. 

Since this book was not designed to be an exhaustive treatise no attempt 
was made to include in it all existing projective differential geometry. Cer- 
tainly, topics already adequately treated in other books could be some- 
what neglected in this one. So, for example, periodic sequences of Laplace 
and the theorems of permutability receive here only passing mention be- 
cause these subjects are extensively discussed in books by Tzitze*ica and 
Eisenhart. Moreover, certain things could be neglected because they seemed 
to be primarily analytical rather than geometrical in their nature; by way 
of illustration may be cited the calculation of complete systems of invari- 
ants and covariants. 

As to arrangement of material, it is hoped that the order in which topics 
spontaneously occurred to the author may prove to be the natural one. 
There is no simpler theory to begin with than that of curves, to which 
Chapter I is devoted. The theory of ruled surfaces, which occupies Chapter 

II, is the next simplest. The elements of both of these theories are prerequi- 
site for the study of surfaces in ordinary space, which is found in Chapter 

III. The subject of conjugate nets, as developed in Chapter IV, leads easily 
to transformations of surfaces in Chapter V. In Chapter VI some parts of 
these projective considerations are specialized so as to show their connec- 
tions with metric and affine geometry. In Chapter VII the projective theory 
of surfaces in hyperspace is amplified to some extent and is generalized in 
order to introduce varieties of more dimensions than two. Finally, Chapter 
VIII contains a number of miscellaneous topics which it seemed unwise 
to exclude altogether and which are to be regarded as more or less supple- 
mentary. 



viii PREFACE 

Certain mathematical attainments on the part of the reader are prerequi- 
site to understanding this book. Some previous acquaintance with the fun- 
damentals of analytic projective geometry is highly desirable, as familiarity 
with homogeneous coordinates is assumed from the outset. The reader 
should be acquainted with, or have constantly at hand, such a book as 
Graustein's Introduction to Higher Geometry. Moreover, the reader is sup- 
posed on occasion to have some knowledge of differential equations, power 
series, and other portions of analysis and algebra. 

There is a list of exercises at the end of each chapter. These are designed 
to give the reader practice in actually working at problems in geometry. 
Many of them are also intended to point the way to further extensions of 
the theory that may be found in the literature. Some of them contain re- 
sults not previously published. 

No attempt has been made to prepare a complete bibliography. How- 
ever, there is a working bibliography at the end of the text, in which the 
references are of two kinds. Some are to the original memoirs. Others are 
to the literature thought to be most convenient for the reader. It is be- 
lieved that the latter references will not be misleading to a student who 
wishes to follow a subject to its source, but will usually serve to guide him 
eventually to the original publication. 

The author hereby gratefully renders due homage to all the great com- 
pany of geometers living and dead whose published researches have been 
drawn upon for the enrichment of these pages. Having availed himself free- 
ly of the labors of others, he wishes that it were practicable in every instance 
to assign credit to whom credit is due for original discovery. The perfection 
of this difficult task must be left to the historian of Projective Differential 
Geometry, but most especially the author wishes to avoid the appearance 
of claiming credit to himself for any discovery made first by another. 

It is a particular pleasure for the author to acknowledge his debt to Wil- 
czynski for early instruction in Projective Differential Geometry, and for 
the first suggestion in 1918 that this book should be written. The author is 
also happy to thank the Italian geometers Fubini, Bompiani, Terracini, 
Castelnuova, and others for their cordiality and many courtesies extended 
to him while he was studying geometry in Italy as Guggenheim Fellow. To 
his colleague, Professor Bliss, the author is indebted for constant encourage- 
ment, for material assistance, and for wise counsel on many occasions. To 
the incisive and penetrating criticisms of Dr. Mendel are due many im- 
provements in the manuscript. 

ERNEST P. LANE 
UNIVERSITY OF CHICAGO 
April, 1931 



CONTENTS 

HAPTBR PAGE 

I. CURVES .... 1 

Introduction . 1 

' 1. Definition and space of immersion of a curve . . . 2 

2. Transformations. Invariants and covariants ... ... 5 

3. Linear osculants . . ... 9 

, 4. Plane curves . . . . ... 12 

, 5. Cubic curves in ordinary space ... . 16 

6. Curves in ordinary space .... 20 

7. The osculating linear complex of a curve in $ 3 25 

Exercises ... ... .... 27 

II. RULED SURFACES . . . ... . 32 

Introduction 32 

8. Elements of the theory of analytic surfaces 33 

9. Developable surfaces ..... 36 

10. Foundations of the theory of ruled surfaces in S n . 38 

11. The differential equations of a ruled surface in S 3 43 

12. The asymptotic curves and the osculating quadric . . 47 

13. The flecnode curves and the flecnode transformation . . .51 

14. The osculating linear complex of a ruled surface in S 3 . . . 54 

15. Quasi-asymptotic curves ... .... ... 56 

Exercises .... ... 58 

III. SURFACES IN ORDINARY SPACE . ... ... 64 

Introduction .... 64 

16. The differential equations of a surface referred to its asymptotic 

net ... 66 

17. The local coordinate system. Power series expansions .... 70 

18. Quadrics of Darboux . . . . ... 75 

19. Reciprocal congruences 81 

20. The canonical pencils 87 

21. Conjugate nets 94 

22. Hyper geodesies. Union curves and planar systems 100 

23. The transformation of tech .... 104 

24. Pangeodesics and the cone of Segre 106 

25. The tetrahedron of Demoulin 109 

Exercises . . 113 



< CONTENTS 

3HAPTER PAGB 

IV. CONJUGATE NETS ... . . ... 121 

Introduction . .... . ... 121 

26. A surface referred to a conjugate net . . . 122 

27. Surfaces sustaining conjugate nets . . . 124 

28. The transformation of Laplace . . .127 

29. Plane nets .... ... 133 

30. Conjugate nets in ordinary space . . 137 

31. Special classes of conjugate nets in ordinary space . .143 

32. Conjugate and harmonic relations of nets and congruences 150 

33. Polar sequences of Laplace . . .159 
Exercises . . . 162 

V. TRANSFORMATIONS OF SURFACES 167 

Introduction . 167 

34. The fundamental transformation 168 

35. The conies and transformation of Koenigs 173 

36. Pairs of surfaces with their points in one-to-one correspondence 177 

37. Pairs of surfaces in <S 8 ... 183 

38. Point correspondence between two surfaces in different spaces $3 . 186 

39. Quadratic nets and congruences, and the transformation of Ribau- 
cour . . . 192 

40. The correspondence between lines of 7? 3 and points on a hyper- 
quadric in 85 . . 194 

41. Surfaces in ordinary ruled space 198 

42. Fubini's theory of W congruences . 205 

43. W congruences in space R 3 . . 209 
Exercises ... ... 213 

VI. METRIC AND AFFINE APPLICATIONS . . 219 

Introduction . . 219 

44. Metric geometry of surfaces . 220 

45. Spheres and pentaspherical coordinates 224 

46. The correspondence between points of M 3 and points on a hyper- 
quadric in $ 4 . 227 

47. The lines of curvature .... 231 

48. A local trihedron at a point of a surface in M& 237 

49. The transformation between cartesian coordinates and Fubini's nor- 
mal coordinates . . .241 

50. Affine geometry of surfaces 244 

51. Surfaces with indeterminate directrix curves . . . 248 
Exercises .... .... . 251 



CONTENTS xi 

CHAPTER PAGE 

VII. SURFACES AND VARIETIES . . 256 

Introduction ... . . 256 

52. The neighborhoods of a point on a surface, or on a variety . . 256 

53. Hyperplane sections of a surface . . ... . 263 

54. Surfaces immersed in space Si . 266 

55. Varieties which are the loci of linear spaces . . 269 

56. The locus of the tangent planes of a surface .... . 273 

57. Sets of varieties which are loci of linear spaces, with the generators 

in correspondence . . 275 

Exercises . . . . . 281 

VIII. MISCELLANEOUS TOPICS . . 286 

Introduction . 286 

58. Historical remarks 286 

59. The method of Fubini . 289 

60. Comparison of certain coordinate systems 292 

61. Congruences in ordinary space 296 
Exercises ... .303 

BIBLIOGRAPHY ... . 305 

INDEX . . .313 



CHAPTER I 
CURVES 

Introduction. Before beginning to develop systematically the theory of 
projective differential geometry with which this book is concerned it seems 
appropriate to describe here in a general way the nature of this kind of 
geometry, and to comment briefly on the contents of the first chapter. 

It is well known that any plane has the property that the straight line 
joining any two distinct points in it lies entirely in the plane. In general, a 
space of any number n of dimensions which has this property is called a 
linear space and is denoted hereinafter by S n . In such a space a non-singular 
protective transformation can be defined analytically as the linear homo- 
geneous substitution that is represented in homogeneous point coordinates 
by a system of equations of the form 

n+l 

P2/i= 2^ a ii%] (*=1, , w-fl; o> = const.) , 
y-i 

where the determinant of the coefficients a, is different from zero, and p is 
an arbitrary factor of proportionality not zero. There are (n+l) 2 coeffi- 
cients aij of which n(n+2) are independent. 

The projective geometry of a configuration is the theory of those properties 
of the configuration that remain invariant under all projective transforma- 
tions of the space in which the configuration lies. Some examples of pro- 
jective invariants are the straightness of a line, the relation of united posi- 
tion of point and line, and the cross ratio of four points on a line. These can- 
not be changed by a projective transformation. Projective geometry may 
be contrasted with metric geometry which studies properties that remain un- 
changed by rigid motions. For example, the distance between two points 
and the angle between two lines are metric invariants but are not projective 
invariants. 

The differential geometry of a configuration is the theory of the properties 
of the configuration in the neighborhood of a general one of its elements. 
In particular, the differential geometry of a curve is concerned with the 
properties of the curve in the neighborhood of a general one of its points. 
In analytic geometry the tangent line at a point of a curve is customarily 
defined to be the limit of the secant line through this point and a neighbor- 
ing point on the curve as the second point approaches the first along the 

1 



2 PROJECTIVE DIFFERENTIAL GEOMETRY 

curve. This definition of the tangent at a point of a curve illustrates the 
nature of differential geometry in that it requires a knowledge of the curve 
only in the neighborhood of the point and involves a limiting process. These 
features of differential, or infinitesimal, geometry show why this geometry 
employs so extensively the differential calculus. Differential geometry may 
be contrasted with integral geometry which views a configuration as a whole. 
The problem of finding the number of intersections of a straight line and a 
conic in the same plane is a problem of integral geometry, since its solution 
requires a knowledge of the entire line and conic. 

These preliminary remarks enable us to describe quite accurately the 
nature of projective differential geometry in a few words. Projective differ- 
ential geometry is, as the name implies, the theory of the protectively in- 
variant differential properties of geometrical figures. This book is mainly 
devoted to the projective differential geometry of curves and surfaces. 
Such other configurations as may appear play a subordinate role. 

The first chapter is devoted to the projective differential geometry of 
curves. In the first three sections analytic curves are considered in a linear 
space of n dimensions. The foundations are laid for Wilczynski's theory of 
curves, which is based upon a consideration of the invariants and covariants 
of an ordinary linear homogeneous differential equation under a suitably 
chosen group of transformations. We do not elaborate this theory here, 
however, even to the extent of computing a complete system of invariants. 
In the remaining four sections we adopt rather the point of view of Halphen 
who studied the projective differential geometry of curves, first in the plane 
in 1878 and then in ordinary space in 1880, by means of power series ex- 
pansions in non-homogeneous coordinates, the coefficients in the expansion 
at a point of a curve being of course constants when the point is fixed, and 
being absolute projective differential invariants of the curve when the point 
is allowed to vary along the curve. This method has the advantage of being 
direct and simple and is easily coordinated with Wilczynski's method, as 
Wilczynski himself has shown. The ordinary osculants of curves are intro- 
duced, and canonical forms are obtained for the power series expansions 
used. The coordinate systems which lead to these expansions are completely 
characterized geometrically. 

1. Definition and space of immersion of a curve. After the introduction 
of certain notations and conventions, an analytic curve will be defined in 
this section. Then analytic conditions necessary and sufficient that the 
locus just defined may reduce to a fixed point, to a straight line, or to a plane 
curve will be deduced. The idea of the immersion of a curve in a linear space 
will be introduced and the space of immersion of a configuration will be de- 



l] CURVES 3 

fined and illustrated. The final result of this section will be the formulation 
of conditions necessary and sufficient that a curve in a linear space S n may 
be immersed in a linear subspace S k of . 

In a linear space of n dimensions S n let us consider a point P with n+l 
projective homogeneous coordinates xi, . . . , x n +i. It is often convenient 
to speak of the point P as the point x, and to indicate this point by P x . 
Sometimes the set (x\, . . . , x n +i) is thought of as a vector x, and then each of 
the coordinates x\, . . . , x n+ i is a component of this vector. A scalar may be 
regarded as a vector with only one component. 

A curve can be described qualitatively as a one-parameter family, or single 
infinity, of points. More precisely, an analytic* curve can be defined as 
follows. // the coordinates xofa point P are single-valued analytic functions of 
one independent variable t } then the locus of the point P, as t varies over its 
range, is an analytic curve C. Analytic curves are the only curves that will 
be considered in this book, although ordinarily it would be sufficient to 
suppose that the functions involved possess a certain number of derivatives. 
When the coordinates x of the point P are expressed as functions of t by 
equations of the form 

(1) *. = *.(<) (i=l, ...,n+l), 

these are spoken of as the parametric equations of the curve C. If we think 
of x as a general one of the coordinates x*, the subscript i can be dropped, 
and so equations (1) can be replaced by what is called the parametric vector 
equation of the curve C, namely, 

(2) x = x(t) . 

There are various possibilities as to the nature of the locus C just defined. 
First of all, it may reduce to a single fixed point, being, in this case only, not 
a proper curve. In fact, if the point P remains fixed as t varies, the ratios of 
the coordinates x are constants, and hence there exists a scalar function X of 
t, which is not zero and is such that 

\x = a (x } a vectors ; a = const.) . 

Therefore x satisfies the ordinary linear homogeneous differential equation of 
the first order 

(3) 



* Goursat-Hedrick, 1904. 1, p. 407. References are to the Bibliography at the end of 
the text, unless otherwise indicated. The figure following the year indicates, in each 
instance, the order in the list appearing under that year in the Bibliography. 



4 PROJECTIVE DIFFERENTIAL GEOMETRY 

accents indicating differentiation with respect to t. Conversely, let us sup- 
pose that the coordinates x satisfy an equation of the form (3). The general 
solution of this equation is 

X = ae-Sp dt (a = const.) ; 

and n+l particular solutions are obtained by giving n+l values to the 
arbitrary constant a while the exponential function remains the same. 
Therefore the ratios of the coordinates x are constants; so the point P is 
fixed and the locus C reduces to a single fixed point. Consequently we have 
the theorem: 

A necessary and sufficient condition that a point be fixed, when the coordi- 
nates of the point are functions of a single variable , is that these coordinates 
satisfy an ordinary linear homogeneous differential equation of the first order. 

A second possibility as to the nature of the locus C will now be consid- 
ered. If the locus C does not reduce to a single fixed point but is a straight 
line, let P a , Pi be two distinct fixed points on this line, while P x is a variable 
point on it. Then it is possible to express x in the form 



where X, JJL are linearly independent scalar functions of t, and a, b are vector 
constants. Differentiating twice and eliminating a, b by use of a determi- 
nant we find that x is a solution of the differential equation 



x X M 
x' X' / 
x" X" M" 



= 0, 



which, by expanding according to the elements in the first column, can be 
written in the form 

(4) s"+2ptf'+pi& = 0, 

because the coefficient of x" is not zero. Conversely, let us suppose that x 
satisfies an equation of the form (4), but satisfies no equation of the form 
(3), so that the point P x is not fixed. The general solution of the equation 
(4) is a linear combination of two particular linearly independent solu- 
tions X, IJL with constant coefficients a, 6; and n+l particular solutions are 
obtained by giving n-f-1 pairs of values to the arbitrary constants a, b. 
Therefore the locus C is the straight line joining the two fixed points P a , P& 



2] CURVES 5 

whose coordinates are, respectively, the n+l values of the two coefficients 
a, b. Hence follows the theorem: 

A curve (2) is not a fixed point but is a straight line in case the coordinates 
x(t) of a variable point on the curve satisfy an equation of the form (4) but 
satisfy no equation of the form (3). 

Similarly it can be shown that a curve (2) is not a fixed point nor a straight 
line but is a plane curve if, and only if, x satisfies an equation of the form 

(5) a/''+3p 1 3"+3p a s'+p 8 x = , 

the coefficients pi, p^ p$ being scalar functions of t, but does not satisfy an 
equation of the form (4) nor one of the form (3). 

In general, a curve C in a linear space S n is said to be immersed in a linear 
subspace S k (k ^ n) of S n in case C is in S k but is not in a linear subspace Sh 
of S k with h <k. If a curve C is immersed in a linear space S k then S k may be 
called the space of immersion of C. In this sense the space of immersion of a 
configuration is the linear space of least dimensions that contains it. For ex- 
ample, the space of immersion of a proper conic in ordinary space is the 
plane in which the conic lies. 

If a curve C is immersed in a linear subspace Sk of a space *S n then it is 
possible to express the coordinates of a variable point P x on C in the form 



where Xi, . . . , \ k +i are linearly independent scalar functions of t, and a (l) , 
. . . , a ( * +1) ate vector constants each of which has for components the 
coordinates of a fixed point in the space S k . Differentiating k+l times and 
eliminating a (1 ), . . . , a (k + l \ we reach the following conclusion: 

A curve (2) in a linear space S n is immersed in a linear subspace S k (k^.n) 
of S n if, and only if, the coordinates x(t) of a variable point on the curve satisfy 
a linear homogeneous differential equation of order k-\-l and do not satisfy 
such an equation of order less than k+l. 

2. Transformations. Invariants and covariants. The contents of this 
section may be summarized as follows. When the curve C, defined by the 
parametric equations (1), is immersed in the linear space S n , an easy calcu- 
lation leads to an ordinary linear homogeneous differential equation of 
order n+l satisfied by the coordinates x(t) of a variable point on C. This 
equation is called a differential equation of the curve C; and conversely, C 
is called an integral curve of the equation. When the equation is given, the 



6 PROJECTIVE DIFFERENTIAL GEOMETRY 

curve is determined except for a projective transformation; so the equation 
does not define the curve uniquely. Moreover, when the curve is given, the 
equation is not determined uniquely, since an arbitrary transformation of 
the'proportionality factor of the homogeneous coordinates, and an arbitrary 
transformation of the parameter that varies along the curve, can still be 
made without disturbing the integral curves. These considerations lead to 
the definitions of invariants and covariants, which are fundamental in 
Wilczynski's method of studying curves. 

Let us consider a curve C with the parametric equations (1), and let us 
suppose that C is immersed in a space S n . The wronskian* of the coordi- 
nates x can be written in the form 

(x, x',..., xW) 

by writing within parentheses only a typical row of the determinant. This 
wronskian does not vanish identically. For, if it did, the coordinates x 
would be linearly dependent, and the curve C would be in a hyperplane 
$ n _i; this conclusion is contrary to the assumption that the curve C is im- 
mersed in the space S n . Therefore, if the coordinates x are substituted one at 
a time in the differential equation 

(6) x< w+1 > + (n+ l) Pl x^+ ---- + Pn+ix = Q , 

the resulting n+l linear algebraic equations can be solved uniquely for the 
coefficients p. The equation (6) with its coefficients thus determined is 
said to be a differential equation of the curve C; and conversely, C is called an 
integral curve of (6). 

When the differential equation (6) is given, any n+l linearly independent 
solutions x\, . . . , x n +i of it can be used as homogeneous coordinates of a 
point P x which, as t varies, generates an integral curve C x of equation (6). 
Let us consider the effect of a non-singular projective transformation, 

n+l 

2/= Pa w 3, (i=l, . . . , n+l; a t , = const.) , 



on the curve C x . This transformation associates with the variable point P x 
a variable point P v which, as t varies, generates a curve C v which is also an 
integral curve of equation (6), since the coordinates y\ t . . . , y n +\ may be 
shown to satisfy (6). The theory of linear differential equations teaches us 
that the most general integral curve of equation (6) can be thus obtained as 

* Goursat-Hedrick, 1917. 1, p. 103. 



2] CURVES 7 

a projective transform of any particular one. The general integral curve, 
just as the general projective transformation, depends on n(n-\-2) parame- 
ters. Hence we reach the conclusion: 

In a space S n the o n(n + 2) integral curves of a given differential equation (6) 
are all protectively equivalent. A differential equation (6) defines a curve in the 
space S n except for a projective transformation; and a geometric theory based on 
the equation must be a projective theory. 

The effect of the transformation of proportionality factor , 

(7) s = X(OJ (X scalar 5*0), 

on equation (6) is found by calculating the successive derivatives of x up 
to and including the one of order n+ 1 and then substituting them in (6) and 
collecting the coefficients of the various derivatives of . The result is a 
differential equation for , of the same form as equation (6). The coefficients 
TTi, . . . , 7T n +i of the new equation are found to be given by the formulas 



(8) 



Since x and differ only by a scalar factor of proportionality, the points P x 
and P coincide. Therefore we have the theorem : 

Equation (6) and any equation into which it can be transformed by a trans- 
formation of proportionality factor (7) have the same integral curves. 

It is possible to choose the proportionality factor X so as to simplify the 
differential equation. If, for instance, the function X is a solution of equa- 
tion (6) then 7r n +i = 0. In particular, we may choose X = a?i; this choice of X 
amounts to introducing non-homogeneous coordinates, 1, 2, . . . , n+i by 
placing 



However, if the function X is a solution of the equation 

X'+piX = 0, 
then TTI = O, and the remaining coefficients are given* by the formula 

P k = eSp> d ^C k>i p t ^ e~S* (fc = 2, 3, . . . , n 

i=0 

* Wilczynski, 1906. 1, p. 16. 



8 PROJECTIVE DIFFERENTIAL GEOMETRY 

where it is understood that po = l- In particular, we have 
(9) Pj = p-p!-p{ , P8 = ps 



These functions P are called seminvariants, because they are invariants 
under, or are unchanged by, the transformation (7) but there is still another 
transformation to be considered. That the functions P are in fact absolute 
invariants under the transformation (7) can readily be verified by means of 
the formulas (8) ; for instance, an easy calculation shows that 



The effect of the transformation of parameter, 
(10) u = u(t) ( 

on equation (6) is found by changing the independent variable in (6) from t 
to u. The calculations are straightforward, and will be omitted. The result 
is another equation of the same form as (6) for x but with the independent 
variable u. Three of the coefficients gi, . . . , q n +i of the new equation are 
found to be given by the formulas 



(U) 



Since the transformation (10) is merely a change of parameter from t to u, 
the curve C is not changed. Hence we arrive at the result: 

Equation (6) and any equation into which it can be transformed by a trans- 
formation of parameter (10) have the same integral curves. 

It is now easy to reduce equation (6) to an especially simplified or canoni- 
cal form. The effect of the total transformation (7), (10) on the functions 
Pi and P 2 is found, by using first formulas (8) and then (11), to be given by 
the equations 

(12) 

in which dashes indicate the transformed functions. Therefore the functions 
X and r) can be chosen so that p\ = F 2 = 0. For this purpose it is sufficient to 
choose T\ as a solution of the equation of Riccati, 



3] CURVES 

and then to choose X so that 



The form of equation (6) thus obtained is called the Laguerre-Forsyth canoni- 
cal form. This form is characterized by the conditions p\ = pz = which, by 
the first of equations (9), are equivalent to pi = P 2 = 0. 

We shall now define certain protective invariants and covariants of a curve 
and indicate briefly how they are connected with the geometry of the curve. 
A function of the coefficients of equation (6) , and of their derivatives, which 
is invariant under the total transformation (7), (10), in the sense that it is 
unchanged by this transformation except possibly for a factor depending 
only on the transformation, is called a protective invariant of an integral 
curve C. Every absolutely invariant equation connecting these invariants is 
independent of the analytic representation of the curve C and hence expresses a 
projective geometric property of C; conversely, every such property can be so 
expressed. A function not only of the coefficients and their derivatives but 
also of x and the derivatives of x, which is invariant under the transforma- 
tion (7), (10) in the sense just indicated, is spoken of as a projective covariant 
of the integral curve C. Every such covariant takes n+1 values when the 
n+1 coordinates x are substituted therein; and when these values are inter- 
preted as the coordinates of a point the covariant defines a curve whose points 
are in one-to-one correspondence with the points of the curve C, and which is ob- 
tainable from C by a projective geometric construction. Conversely, every such 
curve can be so represented analytically. Therefore the projective differential 
geometry of a curve can be studied by means of its invariants and covariants. 
The method of studying a curve thus suggested is that which was used by 
Wilczynski (see Exs. 1, 3). 

3. Linear osculants. Let us consider a curve C immersed in a linear space 
of n dimensions S n . A fundamental problem is to find, for each positive 
integer k which is less than n, the linear subspace Sk of S n which, among all 
linear subspaces with dimensions k, approximates the curve C most closely 
at a given point x on C. This subspace, which will be defined more pre- 
cisely later on in this section, is the osculating space S k of the curve C at the 
point x. Since a linear space Sk is determined by fc-f-1 independent points, 
it will be found that the osculating space S k at an ordinary point x of the 
curve C intersects C in fc+1 ' 'consecutive" points at the point x, and that 
the space S k is consequently determined by x and the first k derivatives of 
x. Among these osculants the tangent line, the osculating plane, and the 
osculating hyperplane receive special consideration. 



PROJECTIVE DIFFERENTIAL GEOMETRY 

We begin with the classical definition of the tangent line. The tangent line 
Si at an ordinary point P x of a curve C is the limit of the straight line through 
P x and a neighboring point PI on C as PI approaches P x along C. If the point 
P x in Figure 1 corresponds to a value t of the parameter of the analytic 




FIG. 1 



curve C, and if PI corresponds to a value t+At, then the coordinates x\ of PI 
can be represented by Taylor's expansion as power series of the form 



(13) xi = x+ 

The point defined by the expression 



is on the secant line PiP*, since this expression is a linear combination of x\ 
and x. The limit of this point, namely the point x', is on the tangent line 
of the curve C at the point P x , and is ordinarily distinct from P x , coinciding 
with P x only if P x is fixed or else is a stationary point (cusp) of the curve C. 
So we reach the following result: 

The tangent line at an ordinary point x of a proper analytic curve C is de- 
termined by the points x and x' . 

Since (Xs)' = X'z+X', it follows that the coordinates x can be multiplied 
by such a common factor X that, after the multiplication, the derivative point 
will coincide with any point whatever on the tangent, except the point x itself. 
In fact, if it is desired that the derivative point shall coincide with a point 
x f +hx, it is sufficient to choose X so that X' = h\. 



3] CURVES 11 

Geometers habitually employ elliptical language to express the idea of the 
limiting process employed in the definition of the tangent line. Thus the 
tangent line at an ordinary point P of a curve C is sometimes said to inter- 
sect C in two "consecutive" points at P. And sometimes any line that cuts 
a curve C in two "coincident" points at a point P, whether they are consecu- 
tive or not, is said to be tangent to C at P. For example, if a plane curve 
has an ordinary node, any line in the plane of the curve and passing through 
the node intersects the curve in two coincident points at the node and is 
called a tangent of the curve. Then the two nodal tangents, or double-point 
tangents, are such that each of them cuts the curve in three coincident 
points at the node, two of them being consecutive points on one branch of 
the curve, and one lying on the other branch. 

Let us next consider the osculating plane, defined as follows. The osculat- 
ing plane $2 at an ordinary point P x of a curve C is the limit of the plane de- 
termined by P x and two neighboring points PI, P 2 on C as PI, P 2 independently 
approach P x along C. Just as the tangent line is said to intersect the curve 
C in two consecutive points at P x , so the osculating plane is said to inter- 
sect the curve in three consecutive points at P x . Therefore the osculating 
plane contains the tangent line. The point defined by the expression 



lies in the plane containing the tangent line of C at P x and also containing a 
neighboring point P 2 on C, since this expression is a linear combination of 
x } x f , and x%. The limit of this point, as P 2 approaches P X) is shown by a 
series of the same form as (13) to be the point x", which therefore lies in the 
osculating plane of the curve C at the point P x . The point x" ordinarily 
does not lie oil the tangent line, being on it only if the curve C is a straight 
line or else has an inflexion at the point P x . So we reach the following 
theorem : 

The osculating plane at an ordinary point xofa proper curve C not a straight 
line is determined by the points x, x', x". 

The osculating space Sk at a point of a curve will now be defined precisely. 
The osculating linear space & at an ordinary point P of a curve C immersed 
in space S n (0<k<n) is the limit of the space S k determined by P and k neigh- 
boring points on C, as each of these points independently approaches P along C. 
Briefly, this is the space S k that intersects C in k+l consecutive points at 
P. Continuation of the reasoning used above leads to the following theorem : 

At an ordinary point x of an analytic curve C immersed in a linear space 
S n the osculating linear space S k (Q<k<n) of C is determined by the points 

~ ' r " r (k) 

A. .*/ u/ ......*/ . 



12 PROJECTIVE DIFFERENTIAL GEOMETRY 

The configuration composed of a point z of a curve C and the osculating 
linear spaces Si, S z , . . . , S r of C at this point is called the element E r of 
the curve C at the point x. The element E r not only has as one of its compo- 
nent parts the osculating space S r , but geometrically is situated in this 
osculating space. 

We conclude this section by proving a theorem concerning a one-parame- 
ter family of hyperplanes. The osculating hyperplanes of a curve obviously 
form a one-parameter family. Conversely, it can be shown that the hyper- 
planes of an analytic one-parameter family osculate a curve. For this purpose 
let us consider a hyperplane, 

n+l 

(14) 



whose n+l coordinates are analytic functions of a variable t. Differentiat- 
ing n 1 times we obtain, dropping the limits from the summation signs, 



Solving these nl equations and (14) for the ratios of the coordinates x, 
and indicating a determinant by writing a typical row within parentheses, 
we obtain the parametric vector equation 



of a curve. The osculating hyperplane of this curve can be shown to be the 
hyperplane (14) by verifying that not only the x in equation (15) but also 
the x', . . . , x (n ~V computed* from (15) satisfy equation (14). Thus the 
proof is complete. Incidentally, we note that in particular the planes of a 
one-parameter family in ordinary space osculate a curve, and that the 
straight lines of a one-parameter family in a plane envelop a curve. 

4. Plane curves. The projective differential geometry of plane curves 
was first systematically studied by Halphen in his Paris doctoral thesis of 
1878, and most of the results of this section are due to him. Together with 
the tangent line and the osculating conic the cubics having eight-point 
contact at a point of a plane curve are especially significant. Among these 
cubics the eight-point nodal cubic and the osculating, or nine-point, cubic 
attract attention. Consideration of these osculants leads to a canonical ex- 

* The derivative of a determinant of order n can be expressed as the sum of n such 
determinants by differentiating column at a time and adding results, thus: (x, y, z}' = 
(z', y, z) + (x, y', z)+(x t y, z'}. See Pascal, / determinant (2d ed.; Milano: Ulrico 
Hoepli, 1923), p. 72. 



4] CURVES 13 

pansion of one non-homogeneous coordinate of a point on a plane curve as a 
power series in the other coordinate. The coordinate system for this ex- 
pansion has a purely geometric description, with which this section closes. 

An algebraic plane curve may be defined as the locus* of a point whose co- 
ordinates #i, Xz, xs satisfy the equation that results from setting a poly- 
nomial in these coordinates equal to zero. The order of an algebraic plane 
curve is defined to be the number of points in which any straight line not a 
component of the curve intersects it. This number is equal to the degree 
of the equation of the curve. For example, a straight line, when regarded 
as a plane curve, is of order one. Of all algebraic curves immersed in a plane, 
the conic is the curve of lowest order, namely, two. 

In analytic projective geometry it is shown that the equation of any non- 
composite conic can be written in the form 



by choosing the coordinate system so that the conic passes through the 
vertex (1, 0, 0) of the triangle of reference tangent to the side z 3 = 0, 
through (0, 0, 1) tangent to i = 0, and through the unit point. If non- 
homogeneous coordinates are introduced by the definitions 



(16) x 

the equation of the conic becomes 

(17) y=x*. 

We shall sometimes use homogeneous coordinates, and sometimes non- 
homogeneous, according to convenience, the notation in each case indicating 
clearly which coordinates are being used. 

The equation of any analytic plane curve C can be written in the form of a 
power series expansion, 

(18) y = a Q +aix+a z x*+ 

This series represents the curve C in the neighborhood of the point P whose 
coordinates are 0, OQ, it being understood that x, y are the coordinates of a 
variable point in the neighborhood of P on C, and that the neighborhood is 
so small that the series converges. We shall obtain a canonical form for the 
expression (18) by choosing the coordinate system so that it will be co- 
variantly, or geometrically, connected with the curve C. 

The osculating conic at a point of a plane curve is defined as follows. The 



14 PROJECTIVE DIFFERENTIAL GEOMETRY 

osculating conic at a point P of a plane curve C is the limit of the conic deter- 
mined by P and four neighboring points on C, as these points independently 
approach P along C. In order to find the equation of the osculating conic K 
at the point P(0, a ) of the curve C represented by equation (18), it is 
sufficient to write the most general quadratic equation in x, y and to demand 
that this equation be satisfied by the power series for y in equation (18) 
identically in x as far as the terms in z 4 . Let us suppose that the coordinate 
system has been chosen so that the equation of the osculating conic K has 
the form (17). Then it turns out that we must have a = ai = a 3 = a 4 = 0, 
a 2 =* 1. The equation of the curve C can thus be written in the form 

(19) y = &+aa?+a#P+<*ix r -\ ---- . 

It will be observed that this equation actually coincides with the equation 
(17) of the osculating conic as far as the terms in a; 4 , and that the coordinates 
of the point P are now 0, 0. Clearly, the curve C and its osculating conic K 
have at the point P the same tangent line, y = Q. 

Ordinarily we have a?^0. For, if a = 0, then the conic K hyperosculates 
the curve C at the point P; that is, K has more than five consecutive points in 
common with C at P, and P is a singular point. Such a singular point is 
called a sextactic point. If every point of the curve C is a sextactic point, 
the osculating conic is the same at every point of (7, and the curve C is there- 
fore itself a conic. We shall as a rule exclude sextactic points from further 
consideration hereinafter. 

There is a pencil of cubic curves each of which has eight-point contact 
with the curve C at the point P. The equation of a general one of these < l 
eight-point cubics can be obtained by writing the most general cubic equa- 
tion in x, y and demanding that this equation be satisfied by the power 
series for y in equation (19) identically in x as far as the terms in x 1 . The 
result is 

a(x*+ay*-xy)+a 6 (y-x*)y+h[a(y--x 2 -axy 2 --a 6 y 3 )-a 7 (y-x 2 )y] = Q , 

where h is an arbitrary constant. The only one of these cubics that has a 
node at the point P is found, by demanding that the derivative dy/dx be 
indeterminate at (0, 0), to be the one for which h = 0. Its nodal tangents are 
found, by equating to zero the terms of lowest degree in its equation, to be 
the lines 



If the triangle of reference is chosen so that the latter line is the side x = 0, 
then 08 = and the equation of the eight-point nodal cubic is 

(20) 



4) CURVES 15 

while the equation of the curve C takes the form 

(21) y 



It is known that every nodal cubic has three collinear inflexions. These 
can be found* by making the equation of the cubic homogeneous and solv- 
ing it simultaneously with its hessian, the solution which gives the double 
point being discarded. Thus we find that the inflexions of the cubic (20) lie 
on the line x\ = 0, some one of them lying on each of the lines 



The branch of the eight-point nodal cubic which is tangent to the curve 
C at the point P is found to be represented by the expansion 



and therefore intersects C in seven consecutive points at P. The other 
branch is represented by 



and therefore merely intersects C in one point at P. 

It is furthermore known that all plane cubic curves through eight given 
points have also a ninth point in common f; it follows that all of the eight- 
point cubics, 

(22) x*+ay*-xy+h[a(y-x 2 -axy*)-by(y-x*)] = Q , 

at the point P of the curve C have a ninth point in common. This point, 
which is usually distinct from P, is called J the Halphen point corresponding 
to the point P of the curve C, and its homogeneous coordinates are found, 
by solving equations (20) and (22) simultaneously for x and y, to be 

(a 5 +6 3 , a 3 6, a& 2 ) . 

Since we are supposing a 7^0, this point coincides with the point P in case 
6 = 0, and then P is called a coincidence point. A curve all of whose points 
are coincidence points is called a coincidence curve. 

* Salmon, 1879. 1, p. 59. 
t Ibid., p. 18. 

J Wilczynski, 1906. 1, p. 68. Halphen, 1918. 4, p. 200; Paris thesis of 1878, entitled 
Sur les invariants diff&rentiels. 



16 PROJECTIVE DIFFERENTIAL GEOMETRY 

Among the eight-point cubics there is a nine-point, or osculating, cubic whose 
equation can be obtained by demanding that the power series for y in equa- 
tion (21) satisfy equation (22) identically in x as far as the terms in x*. 
Thus we obtain the condition on h, 



Let us complete the characterization of the coordinate system by choosing 
the unit point as the point, distinct from P, where the osculating conic inter- 
sects the osculating cubic. Since the unit point is already on the osculating 
conic we now merely demand that it be also on the osculating cubic, and 
find that ah = l and hence a 8 = 6-f-2a 2 . Thus the equation of the curve C 
becomes 

(23) y = x*+ax*+bx''+(b+2a' i )x*+ 

and the equation of the osculating cubic is 



So we reach the following conclusion, illustrated by Figure 2. 

The protective differential geometry of a plane curve C in the neighborhood of 
an ordinary point P on C, can be studied by means of the expansion (23), all of 
whose coefficients are absolute invariants of the curve. The covariant coordinate 
system for this expansion has the following geometric description. One vertex 
of the triangle of reference is the point P(l, 0, 0) under consideration. One side, 
Xz = 0, is the tangent line of C at P. The other side, x z = 0, through P is the 
nodal tangent of the eight-point nodal cubic distinct from the tangent of C; 
and the third side, x\ 0, is the line of inflexions of the eight-point nodal cubic. 
The unit point is the point distinct from P common to the osculating conic and 
the osculating cubic at the point P of the curve C. 

5. Cubic curves in ordinary space. In ordinary space S s an algebraic 
surface may be defined as the locus of a point whose coordinates xi, . . . , x* 
satisfy the equation that results when a polynomial in these coordinates is 
set equal to zero. The order of such an algebraic surface is defined to be the 
number of points in which any straight line that does not lie entirely on the 
surface intersects it. This number is equal to the degree of the equation of 
the surface. A plane, when regarded as a surface in space Ss, is of order one. 
Of all algebraic surfaces immersed in ordinary space the quadric is the sur- 
face of lowest order, namely, two. 

An algebraic curve in ordinary space is by definition all, or part, of the 
intersection of two algebraic surfaces, and the order of such a curve is de- 



5] 



CURVES 



17 



fined to be the number of points in which it intersects any plane not con- 
taining a component of it. Of all algebraic curves immersed in ordinary 
space, the curve of lowest order is the twisted cubic. This curve of order 
three will be studied in this section to prepare the way for the theory of 
unspecialized analytic curves in space 83, which is to follow. 

The simplest possible parametric representation of a twisted cubic will 
first of all be deduced from the definition of the curve, and then the deter- 
mination of the cubic by six of its points will be discussed. Considerations 



x.*o, 




FIG. 2 



of the tangent lines and the osculating planes of the cubic will lead to the 
introduction of the osculating conic at a point of the curve, and to the 
definition of the correspondence between points and planes which is known 
as the null system of the twisted cubic. 

In ordinary space a twisted cubic curve is defined to be the residual intersec- 
tion of two non-composite guadric cones that have one, and only one, generator 
in common (see Ex. 16). The parametric equations of such a curve may be 
found in the following way. If a quadric cone has its vertex at the point 
(1, 0, 0, 0) ; if this cone is tangent to the plane 4 = along the line 4=23 = 0, 
and to 22 = along 22 = 23 = 0; and if this cone contains the unit point, then 
its equation is 

(24) x\ 



18 PROJECTIVE DIFFERENTIAL GEOMETRY 

Similarly, if a cone has its vertex at the point (0, 0, 0, 1 ) ; if this cone is 
tangent to the plane zi = along the line zi = Z2 = 0, and to 3 = along 
#3 =#2 = 0; and if this cone contains the unit point, then its equation is 

(25) xl- 0:1X3 = . 

These two cones obviously have in common the generator xz #3 = 0. The 
remainder of their intersection is the twisted cubic whose equations in 
terms of a parameter t are 

(26) si = l, xa = , x a = t 2 , x* = t* . 

Thus the following theorem is proved: 

By suitable choice of coordinate system and parameter, any twisted cubic 
curve in ordinary space can have its parametric equations written in the form 
(26). 

It is not difficult to demonstrate the truth of the following statement. A 
twisted cubic is determined by six points P J; . . . , P 6 no four of which are co- 
planar. For, there is a unique quadric cone with vertex at PI and contain- 
ing P 2 , . . . , PeJ and there is another unique quadric cone with vertex at P 6 
and containing PI, . . . , P 6 . These cones have the line PiP 6 in common, 
and the remainder of their intersection is a unique twisted cubic* through 
the six given points. 

In order to introduce the osculating conic at a point of a twisted cubic 
we begin with the following remarks. The twisted cubic (26) passes through 
the point (1, 0, 0, 0) tangent to the line x s =x* = Q, since the curve has two co- 
incident points in this line. Moreover, the osculating plane of the cubic at this 
point is the plane # 4 = 0, since the curve has three coincident points in this 
plane. The tangent line at the point x of the cubic that corresponds to any 
value of t is known to be determined by the points x, x', the accent indicat- 
ing differentiation with respect to t; therefore this line has the parametric 
equations 



where X is a parameter. This line meets the plane # 4 = in the point whose 
coordinates are given by 



* Snyder and Sisam, 1914. 1, p. 230. 



5] CURVES 19 

The locus of this point, as t varies, is a conic called the osculating conic of the 
cubic at the point (1, 0, 0, 0). The algebraic equations of this conic are 

(27) 4x^-3x1 = z 4 = 

A remarkable property of the osculating planes of a twisted cubic will now 
be explained, and the null system of the cubic will be defined. The osculating 
plane at any point x of the twisted cubic (26) is known to be determined by 
the points x, x', x"; therefore this plane has the equation 



as can be readily verified by means of equations (26) and the equations ob- 
tained therefrom by differentiating twice with respect to t. Through any 
point y of the space /S 3 there pass three of these planes, whose points of 
osculation have parameter values ti, t^ t 3 , connected by the relations 



tl + t 2 + t S = 32/2/2/1 , tit* + tztz + trfi = 32/3/2/1 , 

These three points of osculation determine the plane 



which evidently also passes through the point y. The coordinates of this 
plane are given by the equations 

(28) & = 2/4, 2= -82/3, 3 = 82/2, 4= -2/i. 

So a twisted cuf)ic determines a one-to-one correspondence between the points 
and planes of ordinary space. When a point is given, the corresponding plane 
is the plane containing the three points of osculation of the three osculating 
planes of the cubic that can be drawn through the given point. Corresponding 
point and plane are in united position. The equations (28) of the correspond- 
ence, being linear, show that it is in fact a correlation. Since a correlation in 
ordinary space which is such that corresponding point and plane are in 
united position is customarily called a null system, we make the following 
definition. The correlation (28) is the null system of the twisted cubic (26). 
If non-homogeneous coordinates are introduced by the definitions 



(29) x = xi/xi , y = xt/xi , z = x*/xi , 
the equations of the twisted cubic become 

(30) y = x 2 , z=x* , 



20 PROJECTIVE DIFFERENTIAL GEOMETRY 

and then the equations of the osculating conic are 
(31) 42/-3z 2 =2=0. 

6. Curves in ordinary space. The projective differential geometry of 
curves in ordinary space was first studied systematically by Halphen in a 
memoir published in 1880. With the notable exceptions of the point first 
defined by Sannia in 1926 and now called the point of Sannia, and of the 
osculating conic, which was introduced by Wilczynski in 1905, the contents 
of this section are, for the most part, found in the memoir* of Halphen. 

In ordinary space an analytic curve can be defined by expressing two of 
the non-homogeneous coordinates of a point on the curve as power series in 
the third coordinate. Some of the more interesting osculants at a point of 
such a curve are the tangent line and the osculating plane, the osculating 
twisted cubic, the osculating conic, and the osculating quadric cone. The point 
of Sannia, the point of Halphen, and the principal plane are associated with 
the point of the curve. Consideration of the various osculants and associ- 
ated configurations yields at the close of this section a geometric descrip- 
tion of a coordinate system for which the equations of the curve assume an 
especially simple canonical form. 

In non-homogeneous coordinates the equations of any analytic curve C in 
ordinary space 83 can be written in the form of two power series expansions, 



These series represent the curve C in a sufficiently small neighborhood of the 
point P whose coordinates are 0, a , c . Since a twisted cubic is determined 
by six of its points we adopt the following definition. The osculating twisted 
cubic at the point P of the curve C is the twisted cubic having six-point contact 
with C at P. Let us suppose that the coordinate system has been chosen so 
that the osculating cubic at the point P(0, a , c ) of the curve C has the equa- 
tions (30). Then equations (32) must coincide with equations (30) as far 
as the terms in z 6 ; thus the equations of the curve C become 

(33) t/ = a; 2 +a 6 zH ---- , z = x*+c*x 6 -\ ---- , 

the coordinates of the point P being now 0, 0, 0. The curve C and its osculat- 
ing cubic have at the point P the same tangent line, y = z = 0, and the same 
osculating plane, z=0. The osculating conic (31) of the osculating cubic of 
* Halphen, 1880. 1. 



6] CURVES 21 

the curve C at the point P is sometimes called* the osculating conic of C at P, 
although it has only two-point contact with the curve C and the cubic. 

There is a bundle of quadric surfaces each of which has seven-point con- 
tact with the curve C at the point P. The equation of a general one of these 
oo 2 seven-point quadrics can be obtained by writing the most general 
equation of the second degree in x, y, z and imposing on it the condition 
that it be satisfied by the power series (33) for y and z identically in x as 
far as the terms in z 6 . The result is 



where h, k, I are arbitrary constants. 

Since all quadrics through seven given points are known to have also an 
eighth point in common, f it follows that all of the seven-point quadrics at 
the point P of the curve C have an eighth point in common. This point is 
calledj the point of Sannia corresponding to the point P of the curve C, 
and its homogeneous coordinates are found to be 

(oj+cj, ojce, a 6 4 cj) . 

If c 6 7^0 let us further restrict the coordinate system by imposing on it a 
condition which may at first sight appear rather artificial; precisely, let us 
choose the point of Sannia for the point (1 , 0, 0, 1 ) . Then o<j = 0, Ce = 1 . There- 
fore the equations of the curve C can be written in the form 



(34) y = 

while the equation of the seven-point quadrics reduces to 



(35) h(y-x*)+k(y*-zx)+l(z-xy-z*)=Q . 

It will be shown presently that the coordinate system is essentially deter- 
mined by the conditions that we have imposed, so that all of the coefficients 
of the series (34) are absolute invariants of the curve C. The geometrical 
significance of the hypothesis made above, that Ce^O, will be explained at 
the end of the next section. . 

Among the seven-point quadrics there are oo 1 cones, for which the dis- 
criminant of (35) vanishes, so that h, k, I satisfy the condition 



* Wilczynski, 1905. 1, p. 112. 

f Snyder and Sisam, 1914. 1, p. 167. 

J Fubini and Cech, 1926. 1, p. 43; Sannia, 1926. 6, p. 18. 



22 PROJECTIVE DIFFERENTIAL GEOMETRY 

The only one of these cones that has its vertex at the point P is the one 
for which h = l=Q, as may be verified by making equation (35) homo- 
geneous and then demanding that the four first partial derivatives of the 
left member vanish at (1, 0, 0, 0). This cone, whose equation is 

(36) . 2/ 2 -*a;=0, 

is called the osculating quadric cone at the point P of the curve C. The locus 
of the vertices of all the seven-point cones is a rational curve of the sixth order 
whose parametric equations can be written in the form 

(37) zi = l-3* 3 , x*=t 5 -t*, x s = t* , z 4 =-* 8 , 

where t 2 = h/l. 

There is a second bundle of quadric surfaces associated with a point P 
of a curve C. If a quadric surface passes through the osculating cubic (30) 
at the point P of the curve C, the equation of the quadric has the form 

(38) h(y-x*)+k(y*-zx)+l(z-xy) =0 , 

since the equation of such a quadric must be satisfied by the expressions 
for y and z in equations (30) identically in x. The totality of such quadrics 
is evidently a bundle. Among these o> 2 quadrics there are oo 1 cones, for 
which hk+l 2 = Q. The vertices of these cones lie on the osculating cubic. 
Among these * cones there are only two that have seven-point contact 
with the curve C at the point P; for them l = hk = Q. The one of these for 
which we have l = h = Q is the osculating quadric cone (36) with its vertex 
at the point P. The other, for which I = k = 0, has the equation 



and its vertex is at the point (0, 0, 0, 1). This point is called* the Halphen 
point corresponding to the point P of the curve C. It is the only point distinct 
from P from which the osculating cubic at the point P of the curve C can be 
projected by a quadric cone having seven-point contact with C at P. 

The principal plane at a point of a curve next engages our attention. If 
the curve C and its osculating cubic at a point P are projected from a point 
onto their common osculating plane at P, the projections ordinarily have 
contact of the same order as that of C and the cubic, namely, fifth-order, or 
six-point, contact. We shall prove in the next paragraph that the pro- 
jections have seven-point contact if, and only if, the center of projection is in the 
plane y = 0. This plane is called the principal plane at the point P of the curve 

* Fubini and Cech, 1926. 1, p. 42. 



6] CURVES 23 

C. This result is a special case of the more general theorem* of Halphen, 
that if two curves have at a point contact of order n, there is one plane, 
called the principal plane, containing the common tangent line of the two 
curves at this point, such that the projections of the two curves from any 
point in this plane onto any other plane have contact of order n-\-l. 

Let us proceed to the proof promised in the last paragraph. The equa- 
tions of the straight line joining any point (x, y, z) of the curve C and any 
point (a, #, 7), which we shall suppose not to be in the osculating plane f = 0, 
are 

(39) t = a+(x-a)t , ^/j+fo-fl* , { = 'Y+(z-y)t , 

wherein y, z are given by equations (34) and t is a parameter. The equa- 
tions of the projection of C from (a, 0, 7) onto the plane t = are found by 
using t= y/(z 7) in equations (39). The result is 



(40) 

Inverting! the first of these series to obtain z as a power series in , we find 



and substituting the result in the second of (40), we obtain the equation of 
the projection of the curve C onto its osculating plane, f = 0, namely 

(41) 77 = f 2 -^ 3 /7-h2afV7-(7H-3a0)f/7 2 +(7a 2 +2^-/37)f 6 /7 2 + 

If we suppose that y, z, in equations (39) are given by (30) instead of by 
(34), and repeat the operations just outlined, we arrive at the equation of 
the projection of the osculating cubic onto the osculating plane; this equa- 
tion is found to be the same, to terms of order six, as equation (41) except 
that in the coefficient of 8 the term 7 is missing. Therefore the projections 
have seven-point contact if, and only if, = 0, since 75^0; in this case the 
center of projection lies in the plane y = 0, as was to be proved. 

The principal point of the tangent line at a point of a curve has the fol- 
lowing definition. The point corresponding to the principal plane, y = 0, in the 
null system of the osculating cubic isl the principal point of the tangent. Its 

* Halphen, 1880. 1, p. 25; 1918. 4, pp. 375-76. 
t Goursat-Hedrick, 1904. 1, p. 406. 
t Wilczynski, 1906. 1, p. 271. 



24 



PROJECTIVE DIFFERENTIAL GEOMETRY 



homogeneous coordinates are found by use of equations (28) to be (0, 1, 0, 0,) ; 
so the principal point lies on the tangent line, XB = x* = 0. 

The coordinate system for the expansions (34) can now be visualized 
as in Figure 3 and described geometrically as follows. The vertex (1, 0, 0, 0) 
of the tetrahedron of reference is the point P of the curve under considera- 



tangent 



cubic 




FIG. 3 



bion, and the edge x 9 =x*=Q is the tangent of the curve C at the point P. 
The face x = is the osculating plane, and the face 3 = is the principal 
plane, of C at P. The vertex (0, 1, 0, 0) is the principal point of the tangent. 
The edge #2 = 4 = is the polar line of the principal point with respect to the 
osculating conic (27) of C at P; and the vertex (0, 0, 1, 0) is the point dis- 
tinct from P where this line meets the osculating conic. The vertex 
(0, 0, 0, 1) is the Halphen point of C at P. Thus the tetrahedron is char- 



7] CURVES 25 

acterized. The unit point is one of the three points where the osculating 
cubic intersects the plane x\ #4 = determined by the vertex (0, 0, 1, 0), 
the point of Sannia (1, 0, 0, 1), and the principal point (0, 1, 0, 0). 

7. The osculating linear complex of a curve in 83. We shall frequently 
have occasion hereinafter to consider certain configurations of lines. It is 
known that all the straight lines of ordinary space 3 form a four-parameter 
family. A three-parameter family of these lines is called a line complex, or 
simply a complex; a two-parameter family is called a congruence; and a one- 
parameter family, a ruled surface. 

After the pluckerian homogeneous coordinates of a line in ordinary space 
are introduced early in this section it is easy to define a line complex ana- 
lytically and in particular a linear line complex. The null system of a linear 
complex appears next for brief consideration. The osculating linear complex 
at a point of a curve is then defined and its equation, referred to the co- 
ordinate system of the preceding section, is calculated. Finally, the condi- 
tions are obtained that the osculating linear complex at a point of a curve 
may hyperosculate the curve at the point, and that a curve may belong to 
a linear complex. 

We proceed to introduce the pluckerian coordinates of a line. If y\, . . . , i/ 4 
and 21, . . . , 04 are homogeneous coordinates of two distinct points P v and P t 
on a line I in ordinary space, then the pluckerian homogeneous coordinates of I 
are defined to be the six numbers coi 2 , wis, WH, w 23 , ^42, ^34 given by 

(42) o>ik = y,Zk-ykZi (i, & = 1, . . . , 4) . 



It is cleat" that only the ratios of these coordinates are determined when the 
points y and z are given; and it may be verified that, if any other pair of 
points on the line I is used in place of y, z, essentially the same line co- 
ordinates result. Moreover, these coordinates satisfy the equation 

(43) <*>1 2 W34 + Wi3C042 + COi4a)23 = , 

as can be verified by direct substitution. Conversely, in treatises on line 
geometry it is shown that any six numbers satisfying this relation can be 
regarded as the coordinates of a line. 

We next state some fundamental definitions. The locus of a line whose 
coordinates satisfy a homogeneous equation with constant coefficients is a com- 
plex. If the equation is linear, of the form 

(44) 



0342/1 0142/3+0132/4 , 

0422/1 + 0142/2 0122/4 , 

y i 0132/2+0122/3 



26 PROJECTIVE DIFFERENTIAL GEOMETRY 

the complex is called a linear complex. Substituting in this equation the ex- 
pressions for the coordinates w^ given in (42), and collecting coefficients of z, 
we obtain an equation of the form Sz = whose coefficients are defined, 
except for a non-vanishing proportionality factor p, by the equations 



(45) 



If the point P v is held fixed and P z is allowed to vary, we see that all of the 
lines of a linear complex that pass through a fixed point lie in a fixed plane 
which passes through the point, and therefore form a flat pencil with center at 
the point. So a linear complex determines a correspondence between the 
points and planes of ordinary space, which is, in fact, a correlation with 
corresponding point and plane in united position. This correlation deter- 
mined by a linear complex and represented analytically by equations (45) 
is called the null system of the complex. 

Since equation (44) contains five essential constants it follows that a 
linear complex is determined by five of its lines. Hence we make the fol- 
lowing definition. The osculating linear complex at a point P of a curve C is 
the limit of the complex determined by the tangent of C at P and the tangents at 
four neighboring points of (7, as each of these points independently approaches 
P along C. In order to find the equation of the osculating linear complex of 
the curve (34) at the point (0, 0, 0), we observe that, when non-homo- 
geneous coordinates are introduced by the definitions (29), the coordinates 
of the line joining the points (x, y, z) and (, 77, f) are given by 



/^X 

(46) 

The parametric equations of the tangent line of the curve (34) at the point 
x z are 



where X is the parameter and accents denote differentiation with respect to 
x. The line coordinates of this tangent are therefore given by 

f Wl2=l , Wi 8 = 2/' > 0>i4 = 2' , 

' ' '' 



7] CURVES 27 

Using the series (34) and the expressions obtained therefrom for y' t z', we 
get, to terms of the fourth order, 



(48) 



and demanding that equation (44) be satisfied by these expressions identi- 
cally in x as far as the terms in s 4 , we find the equation of the osculating linear 
complex at the point (0, 0, 0) of the curve (34), namely, 

(49) 0>i4- 3^23 = . 

For this complex equations (45) reduce to (28). Therefore we have the 
theorem: 

The null system of the osculating linear complex at a point P of a curve C is 
the same as the null system of the osculating twisted cubic of C at P. 

We conclude by explaining the geometrical significance of the hypothesis 
cer^O, which was made in the last section. If in computing the equation 
of the osculating linear complex we had used equations (33) and the tetra- 
hedron of reference described in connection therewith, instead of (34), the 
equation of this complex would still have turned out to be precisely (49). 
Computing the left member of (49) to terms of the fifth degree we find 



Therefore a point at which Ce = is a point at which the osculating linear com- 
plex hyperosculates the curve. Such singular points were excluded in deriving 
equations (34). If every point of a curve is such a point, the osculating 
linear complex is the same at every point of the curve, and the curve be- 
longs to a linear complex, in the sense that its tangents are lines of the 
complex. 

EXERCISES 

1. The functions 0* and 8 of the coefficients of equation (5) and their derivatives, 
which are defined by placing 



are relative invariants of weights 3 and 8 respectively under the transformation 
(10), and are absolute seminvariants. WILCZYNSKI, 1906. 1, p. 59 



28 PROJECTIVE DIFFERENTIAL GEOMETRY 

2. If equation (5) is in its Laguerre-Forsyth canonical form, and if a point 
y&+y& f +y&" is said to have local coordinates 3/1, y 2 , 2/s, show that the local coordi- 
nates of a point near the point P on the curve C are represented by 



Hence show that the local equation of the osculating conic of C at P is y\ 2y\ys = 0. 

WILCZYNSKI, 1906. 1, p. 62 

3. Use the Laguerre-Forsyth canonical form of equation (5) to prove that if 
03=0, the integral curves of (5) are all conies. WILCZYNSKI, 1906. 1, p. 90 

4. Use the result of Exercise 1 and the first of equations (12) forn = 2 to obtain the 
Halphen canonical form of equation (5), for which pi = 0, 0s=l. What is the most 
general transformation (7), (10) preserving this canonical form? Prove that for 
this form the points x' y x" are co variant, and describe them geometrically. 

5. Show that by a suitable transformation of unit point equation (23) becomes 



where the coefficients A, B are given by 



Describe the new unit point geometrically. 

6. In the notation of equation (23) the cross ratio of the tangent of the curve at 
the point P, the other double-point tangent of the eight-point nodal cubic, the line 
from P to the Halphen point, and the line from P to the unit point is 6/a 2 . 

7. Among the seven-point quadrics (35) there is a pencil of eight-point quadrics 
whose equation is 

h[y-x*+a(y*-zx)]+l[z-xy~z*+c(y*-zx)] = Q. 

The one of these eight-point quadrics for which hQ is the only one that contains 
the line 2/= s =0; and is the only one to which the line #=z=0 is tangent. The 
eight-point quadric for which Z = is the only one containing the line x = y = Q; is 
the only one which meets the line Xi =2= in points separating the points (0, 1, 0, 0) 
and (0, 0, 1,0) harmonically; and is the only one containing the osculating cubic 
(30). For the osculating, or nine-point, quadric the ratio h/l satisfies the condition 
h(b ac)+l(d a c 2 ) = 0; and for the four eight-point cones this ratio satisfies the 
condition 



CURVES 29 

8. The twisted cubics cut on the osculating quadric cone (36) by the l cones 
yx*+hyg=Q are the i five-point cubics of the curve C at the point P; their para- 
metric equations can be written in the form 



All of these cubics have the same osculating conic at P, and have the same null 
system. 

9. The coordinates of the osculating plane at the point P of a five-point cubic 
defined in Exercise 8 are given by 



and satisfy the equations 



The coordinates of the osculating plane of the curve (34) at P are given by 
&=**+ - , 2= -3a*- 



Prove that every five-point cubic is also a five-plane cubic of C at P, and that the 
five-point cubic for which /&=2 is a six-plane cubic. 

FUBINI and CECH, 1926. 1, p. 42 

10. The osculating linear complex of the osculating twisted cubic at a point P 
of a curve C in 3 is the same as the osculating linear complex of C at P. 

11. Defining an anharmonic plane curve to be a curve for which, in the notation 
of Exercise 4, 

const., 



use the Halphen canonical form of equation (5) mentioned in Exercise 4 to show 
that an anharmonic curve admits a one-parameter group of protective transforma- 
tions into itself. Show that for each anharmonic curve there exists a triangle such 
that the cross ratio of a point P on the curve and the three points where the tangent 
at P meets the triangle is constant. WILCZYNSKI, 1906. 1, p. 86 

12. At a coincidence point of a plane curve the osculating cubic of the curve has 
a node. One of the branches of the cubic has eight-point contact with the curve at 
the point, and the other branch merely passes through the point. 

HALPHEN, 1918. 4, p. 207 and p. 205 

13. If a curve C in ordinary space belongs to a linear complex, the tangent line 
at a variable point of C intersects the osculating plane at a fixed point P of C in a 
point whose locus is a plane curve with a sextactic point at P. 

HALPHEN, 1880. 1; 1918. 4, pp. 430-31 



30 PROJECTIVE DIFFERENTIAL GEOMETRY 

14. In the notation of equations (34), the equation of the cubic cone having its 
vertex at a point P of a curve C, having the tangent line of C at P for double genera- 
tor, and having eleven-point contact with C at P, is 



This cone has twelve-point contact in case d2a c 2 =0. 

HALPHEN, 1880. 1; 1918. 4, p. 432 

15. In a plane consider an integral curve C of equation (5) and define a general 
one of the three line coordinates of the tangent at a point & of C by the formula 



Then show that satisfies the Lagrange adjoint of (5), namely 



Generalize this result for a curve in space S n . 

16. In ordinary space two non-singular quadric surfaces having one, and only 
one, generator in common intersect elsewhere in a twisted cubic. 

17. Prove that the differential equation of a coincidence curve can be reduced to 
the form x'"-\-x = Q. Hence show that every coincidence curve is protectively 
equivalent to a logarithmic spiral which intersects all of its radii at an angle of 30. 

WILCZYNSKI, 1906. 1, p. 69 

18. If two plane curves have at a point P contact of order fc 1, and if non-homo- 
geneous protective coordinates are chosen so that the expansions representing these 
two curves in the neighborhood of P are, respectively, yax k -\- , 2/=6x*+ 
(a^b), then the ratio a/b is a projective invariant of the curves, with the following 
interpretation. Consider a transversal straight line cutting the two curves in points 
R, S, near P, and cutting the common tangent at P in a point T, also near P. Let 
M be any fourth point on the transversal. Then the limit, as T approaches P, of the 
cross ratio (RSTM) is a/b, provided that the limit of the transversal is not the com- 
mon tangent, and also provided that the limit of the point M is not the point P. 

SEGRE, 1897. 1, p. 170 

19. In Exercise 18 let k =2, and calculate the cross ratio (RSTM) as a power 
series in the coordinate not zero of the point T. The term of order zero is a/b and is 
independent of the limit of the transversal and of the point M. The term of the first 
order depends on the limit of the transversal but not on the limit of M . This term 
vanishes if the limit of the transversal is the harmonic conjugate of the common 
tangent with respect to the two lines projecting from P the two points of intersection 
of an arbitrary conic having four-point contact with one curve at P and another 
conic having similar contact with the other curve. The term of the second order 



CURVES 31 

depends on the limit of the point M , when the limit of the transversal is fixed in the 
way just described. The point in which this limit line touches the curve enveloped 
by the transversal is protectively related to the point M that makes the term of the 
second order vanish. The double points of the projectivity thus determined on the 
limit line, one of which is P, and the two points in which this line meets the osculat- 
ing conies of the two curves at P, have a cross ratio which is a/b. Discuss the situa- 
tion when a = b. BOMPIANI, 1926. 8 

20. If two curves in ordinary space have at a point P contact of order n, and if 
their principal plane is different from their common osculating plane at P, there 
exists a line through P in the principal plane such that the projections of the two 
curves from any point on this line onto any plane have contact of order n-f-2. On 
this line, called the principal line, there is a point, called the principal point, such that 
the projections of the curves from this point have contact of order n-f 3. Applying 
this result to a curve and its osculating twisted cubic, let = and calculate the 
expansion (41) to terms of the eighth order; thus show that the equations of the 
principal line are /3 = 2a-|-a7 = 0, and that the coordinates of the principal point are 

-ac), 0, l/(6-ac)] . 

BOMPIANI, 1926. 9 



CHAPTER II 
RULED SURFACES 

Introduction. Just as a curve can be thought of as the path of a moving 
point, so a ruled surface can be regarded as the locus of a moving straight 
line. A ruled surface can also be described as a one-parameter family, or 
single infinity, of straight lines, which are called the generators of the surface. 
The essential characteristic property of a ruled surface is that through each 
point of the surface there passes at least one straight line that lies entirely on the 
surface. A quadric cone is an example of a ruled surface; and a non-singular 
quadric surface in ordinary space is an example, in fact the only example, of 
a surface that is ruled in two ways so that there are precisely two generators 
through every point of the surface. 

In order to avoid needless repetition it seems desirable to present in the 
opening section of this chapter the exact definition of an analytic surface, 
and to develop the elements of the projective theory of such surfaces. The 
next section is devoted to the special class of ruled surfaces called develop- 
able surfaces. The foundations are laid in the following section for the pro- 
jective theory of general ruled surfaces in a linear space of n dimensions. 

Wilczynski in 1906 published in book form his projective theory of curves 
and of ruled surfaces in ordinary space, to which during the preceding five 
years he had devoted about ten memoirs, published mostly in the Transac- 
tions of the American Mathematical Society. This theory was a significant 
contribution to projective differential geometry. With it Wilczynski gave 
to the world a new method in geometry and established himself as the 
founder of a new school of geometers. 

Wilczynski's theory of ruled surfaces is based upon a consideration of the 
invariants and covariants of a system of two ordinary linear homogeneous 
differential equations of the second order in two dependent variables, under 
a suitably chosen group of transformations. We shall employ this analytic 
basis in four sections devoted to ruled surfaces in ordinary space, namely, 
Sections 11-14. In these we shall not elaborate the theory of complete sys- 
tems of invariants and covariants. However, we shall preserve Wil- 
czynski's notations and include some of his most fundamental geometrical 
results. 

In the concluding section of this chapter a brief account is given of one 
of Bompiani's contributions to the theory of ruled surfaces in hyperspace. 

32 



8] RULED SURFACES 33 

This is the theory of quasi-asymptotic curves, which are generalizations of 
ordinary asymptotic curves. 

8. Elements of the theory of analytic surfaces. The purpose of this sec- 
tion is to develop some of the elementary parts of the theory of general 
analytic surfaces, which will be used later on in discussing ruled surfaces. 
Parametric curves and curvilinear coordinates on a surface, and curvilinear 
representations of families and nets of curves on a surface, are introduced. 
Consideration of the tangent line and the osculating plane at a point of a 
curve on a surface leads to the definition of the tangent plane at a point of a 
surface, and to the definition of asymptotic curves. 

A surface can be described as a two-parameter family of points. More 
precisely, an analytic surface may be defined as follows. // the n+l homo- 
geneous coordinates x of a point P x in a linear space of n dimensions S n are 
given as single-valued analytic functions of two independent variables u, v, by 
equations of the form 

(1) x = x(u,v), 

then the locus of P x as u, v vary is an analytic surface S. Equation (1) is 
spoken of as the parametric vector equation of the surface S. If the coordi- 
nates x satisfy no linear homogeneous partial differential equation of the 
first order of the form 

(2) ax+bx,+cx = , 

in which subscripts indicate differentiation, and the coefficients a, 6, c are 
scalar functions of u, v not all zero, then the surface S is called a proper 
analytic surface. In fact, if the coordinates x of a point P x are given as 
analytic functions of u, v and are supposed to satisfy an equation of the 
form (2), then integration of this equation would show that these coordi- 
nates could be expressed by the formula 

*< = */.() (i-l,...,n+l), 

wherein <? is a function of u, v which is a particular solution of equation (2), 
and t is a function of u, v which is a particular solution of the equation at u + 
bt v =0, while the/ are functions of t, as is indicated by the notation. Then, 
since the ratios of the coordinates z would be functions of the single vari- 
able t, the locus of the point P x would reduce to a curve, which might further 
degenerate into a single fixed point (see Ex. 1 of Chap. III). In what fol- 
lows, when we speak of a surface this will be understood to be a proper 
analytic surface unless the contrary is indicated. Moreover, singular points 



34 PROJECTIVE DIFFERENTIAL GEOMETRY 

where equation (2) may happen to be satisfied conditionally on such a sur- 
face will be avoided. 

If the parameter v is fixed while u varies, the locus of the point P x is a 
curve, called a u-curve, and sometimes denoted by C u , on a surface S. There 
are < 1 w-curves on S, along each of which v const, and therefore dv 0. 
Similarly there are i v-curves on the surface S, along each of which u = 
const, and du = Q. These curves are called the parametric curves on the sur- 
face S, and their tangents are called respectively u-tangents and v-tangents, 
and collectively the parametric tangents. Two one-parameter families of 
curves on a surface S are said to form a net in case through each point of S there 
passes just one curve of each family, the two tangents of the curves at the point 
being distinct. We shall suppose from now on that, on the portion of the 
surface S under consideration, the parametric curves form a net, which will 
be called the parametric net. 

Each pair of values of the parameters u, v locates a point P x on a surface 
S. In a region on S where points P x and pairs of values of u, v are in con- 
tinuous one-to-one correspondence, u, v are correctly called coordinates of P x . 
It is customary to designate these coordinates as curvilinear coordinates to 
distinguish them from the projective homogeneous coordinates x. 

Any curve on a surface S can be represented by a curvilinear equation, i.e., 
an equation in curvilinear coordinates, of the form <p(u, v) = 0, and can also 
be represented parametrically by placing 

(3) u = u(f) , v = v(t) . 

Any oo 1 curves, except the curves u const., on the surface S can be repre- 
sented by a curvilinear differential equation of the form 

(4) dv X (u, v)du = Q ; 

and any net of curves on $ has a curvilinear differential equation of the form 

(5) Adu*+2Bdudv+ C(& 2 = (AC-flVO) . 

In particular, the differential equation of the parametric net is dudv = 0. 

It will now be proved that at an ordinary point on a surface the tangent 
lines of all the curves on the surface through the point form a flat pencil with 
its center at the point. The tangent line at a point P x of a curve C with equa- 
tions (3) on a surface S is determined, as in Figure 4, by P x and the point a;', 
where 

(6) x'=x u 



8] RULED SURFACES 35 

accents indicating differentiation with respect to t. If the point P x remains 
fixed while the curve C through P x varies in all possible ways on the surface 
S, then x u , x v remain fixed while the derivative dv/du varies. The locus of 
the point x' is therefore the line joining the points x u , x v , the first of which 
lies on the w-tangent of S at P x , and the second on the v-tangent. Therefore 
the locus of the tangent line xx' is a flat pencil lying in the plane of the points 
x, x u , x v and with its center at the point P x , as was to be proved. 




FIG. 4 

We are now ready to state the classical definition of the tangent plane. 
The tangpnt plane at a point of a surface is the plane containing the tangent 
lines at the point of all the curves on the surface through the point. The tangent 
plane at a point x of a surface S referred to parameters Uj v is determined by 
the points x, x u , x v . If the surface S is in ordinary space S 9 , the equation of 
the tangent plane can be written in the determinantal form 

(7) (X, x,x u , z v )=0, 

in which the point X is variable on the plane. 

Since the position of the tangent line at a point P x of a curve C on a sur- 
face S is determined by the derivative dv/du calculated from the curvi- 
linear representation of C, this derivative is called the direction of the curve 
C at the point P x . 

An asymptotic curve on a surface may be defined projectively as follows. 
A curve on a surface is an asymptotic curve in case at each of its points its 
osculating plane coincides with the tangent plane of the surface at the point. 



36 PROJECTIVE DIFFERENTIAL GEOMETRY 

The osculating plane at a point P x of a curve (3) on a surface S is deter- 
mined by x, x', x", where x' is given by equation (6) and x" by 

(8) x" = x uu u' 2 +2x uv u'v'+x vv v' 2 +x u u"+x v v' ' . 

If the surface S is in ordinary space, the curve (3) is an asymptotic curve in 
case not only x, x' but also x" satisfy equation (7) when substituted therein 
in place of X. Substitution of x" for X in (7), and reduction of the result, 
lead to the following theorem. 

The curvilinear differential equation of the asymptotic curves on a surface 
(1) in ordinary space is 

(9) Ldu z +2Mdudv+Ndv* = , 

where the coefficients L, M, N are the determinants of the fourth order defined 
by 

(10) L= (x uu , x, x u , x v ) , M = (x uv , x, x u , x v ) , N = (x vv , x, x u , x v ) . 

We conclude this section with a few remarks pertaining to asymptotic 
curves. Usually the asymptotic curves on a surface in ordinary space form a 
net. The exceptional case will be considered at the close of the next section. 

The tangent line at a point of a straight line is the line itself; the osculating 
plane at a point of a straight line is indeterminate. Every straight line on a 
surface is included among the asymptotic curves on the surface, since the tan- 
gent plane of the surface at a point of the straight line is an osculating plane 
of the straight line. 

The reason why the asymptotic curves on a surface are called by this 
name is a historical one. They were first defined* as those curves whose 
tangents at each point of the surface are the asymptotes of the conic now 
commonly called the Dupin indicatrix of the surface at the point. This 
property is, however, not a protective property. 

9. Developable surfaces. The reason why developable surfaces are so 
named is that in ordinary space analytic developable surfaces are the only 
analytic surfaces that can be developed upon, or applied to, a plane. This 
means that it is possible to set up a one-to-one correspondence between the 
points of an analytic developable surface in ordinary space and the points 
of a plane such that corresponding curves between corresponding points 
have equal lengths, and no other analytic surfaces in ordinary space have 
this property. Since this definition is not of a projective nature, it is neces- 

* Dupin, 1813. 1, p. 51. 



9] 



RULED SURFACES 



37 



sary to substitute for it a purely projective definition of developable sur- 
faces before admitting this important class of ruled surfaces to the domain 
of projective differential geometry. We proceed to do so in this section. We 
then set up the parametric vector equation of a developable surface and 
study some of the fundamental properties of developable surfaces. 

In space S n a developable surface is defined to be the locus of the tangent lines 
of a curve, and is sometimes called simply a developable. The tangents are 
known as the generators of the developable; the curve is called the cuspidal 
edge, or edge of regression, of the developable; and the point of contact of a 
generator with the cuspidal edge is designated as the focal point of the gener- 
ator. 

Certain special cases should be mentioned here. If the edge of regression 
is a plane curve, the developable is obviously all or part of the plane of the 
curve. If the edge is a 
straight line, the developa- 
ble evidently reduces to this 
line and is not a proper sur- 
face. If the edge reduces to 
a fixed point, a little medi- 
tation shows that the de- 
velopable is to be thought 
of as a cone with its vertex 
at the point. 

The next problem is to 
write the parametric vector 
equation of a developable. 
Let the parametric vector 
equation of the edge of re- 
gression C v of a developable 

be y = y(t), as indicated in Figure 5. This curve is the locus of a point P v ; and 
any point P x , except P v , on the tangent of C v at P y is defined by placing 




FIG. 5 



(11) 



(u scalar) . 



If the parameter t is fixed while u varies, the locus of the point P x is the 
tangent line ; but if u, t both vary, the locus of P x is the developable surface. 
Hence we reach the conclusion : 

Equation (11) is the parametric vector equation of a developable surface. 

Certain properties of developable surfaces merit attention. The tangent 
plane at a point P x of the developable (11) is determined by the points x, x u , 
x t . If the derivatives x u , x t are calculated it is easy to see that the tangent 
plane may also be regarded as determined by the points y, y', y". But these 



38 PROJECTIVE DIFFERENTIAL GEOMETRY 

points determine the osculating plane of the edge of regression at the 
point P v . Thus we prove the theorem : 

The tangent plane at every point of a generator of a developable surface is 
the same plane, and is the osculating plane of the edge of regression of the de- 
velopable at the focal point of the generator. 

It follows from this theorem that a developable surface has only < l tangent 
planes instead of the usual 2 . Conversely, in ordinary space the planes of 
any analytic one-parameter family osculate a curve, according to the con- 
clusion of Section 3, and therefore envelop a developable surface. 

Using the expression for x given in equation (11), let us replace v by Z in 
equations (9), (10), and calculate the determinants L, M, N defined by (10). 
We find L = M = 0, N=(y f ", y", y', y). If # = 0, the curve C v is a plane 
curve, and conversely. Let us suppose TWO. Then (9) shows that the 
asymptotic curves on a developable surface immersed in ordinary space consist 
only of the generators of the surface. Consequently these curves do not form 
a net. Conversely, it can be shown that if the asymptotic curves on a sur- 
face S immersed in ordinary space form only a one-parameter family, then S is a 
developable surface. For, in the first place, we have LN M 2 = 0. If the fami- 
ly of asymptotic curves is taken for the ^-curves on the surface S, then 
L = M = 0, and consequently x satisfies two equations of the form 



(12) x uu = px+ax u +(3x v , x uv =*cx+ax u +bx v , 

whose coefficients are scalar functions of u, v. The coordinates of the tan- 
gent plane (7) can be defined by 

{=(, x u , O ; 

and in the presence of equations (12) differentiation yields u= (<*+&)?. 
Therefore the tangent plane at every point of a w-curve is the same plane, 
the ratios of the coordinates being independent of u. Consequently the 
surface S is developable. Thus the following theorem is proved. 

The asymptotic curves on a surface immersed in ordinary space fail to form 
a net if, and only if, the surface is developable. 

10. Foundations of the theory of ruled surfaces in S n . The parametric 
vector equation of a general ruled surface in a linear space of n dimensions 
can be written very simply by the aid of two curves with their points in 
one-to-one correspondence. After writing this equation and deducing the 
most elementary properties appertaining to the tangent planes of a ruled 
surface, the remainder of this section will be taken up with an introduction 



iio] 



RULED SURFACES 



39 



to Bompiani's theory of the various osculating linear spaces that can be 
associated with an element of a curve on a ruled surface. 

A ruled surface is a one-parameter family of straight lines. Therefore 
every developable surface is ruled. But not every ruled surface is developable. 
For example, among the non-developable ruled surfaces in ordinary space is 
the quadric x^ z i4 = 0, which may also be defined by the parametric 
equations 

and which has on it the two distinct one-parameter families of straight lines 
X2 = uxi, xt uxs and x$~vx\, 4 = ^2. These lines can be shown by equa- 
tions (9), (10) to be the asymptotic curves on the quadric, and they consti- 
tute the parametric net for the representation used in equations (13). 

In order to write the parametric vector equation of a general ruled sur- 
face, let us consider in space S n two curves C v , C z as in Figure 6. Let us 




FIG. 6 



refer these curves to the same parameter , and let us join a pair of points 
P y , P z (supposed distinct) corresponding to the same value of t by a straight 
line l vz . The locus of l yz , as t varies, is a ruled surface R, and every ruled 
surface can be generated in this way. Any point P x , except P VJ on l vt is 
defined by placing 

(14) x = z+uy (u scalar) . 



If t is fixed while u varies, the locus of the point P x is a generator of the ruled 
surface R. If u } t both vary, the locus of P x is the surface R. Hence the use- 
ful result: 

Equation (14) is the parametric vector equation of a general ruled surface. 



40 PROJECTIVE DIFFERENTIAL GEOMETRY 

It may be observed that the ruled surface R is developable if z=y', and 
is a cone if z = const. The case that the surface reduces to a single fixed line 
is excluded. The curves C V) C z will be called director curves of the ruled 
surface R. It is clear that any two distinct curves on a ruled surface may 
be taken as the director curves. 

We shall next show that if there are two distinct curves on a ruled surface 
such that their tangent lines at the points where they cross each generator are co- 
planar , then the ruled surface is developable. For, if the two curves are taken 
as the director curves C v , C z , then the points y, z, y', z' are coplanar and 
there exists a relation of the form 



(15) 

where a, 6, c, d are scalar functions of t. This equation is equivalent to 

(cy+dz)'=(c'-a)y+(d'-b)z. 

Hence, when t varies, the point cy+dz describes a curve to which the genera- 
tor l yz is tangent. Therefore all the generators of the surface are the tan- 
gents of a curve, and the surface is developable, as was to be proved. 

The tangent plane at a point P x of a ruled surface (14) is determined by 
x, x u , x t and therefore by y, z, z'+uy'. Consequently the tangent plane at a 
point of a ruled surface contains the generator of the surface through the point, 
as is also evident geometrically. Conversely, every plane through a generator 
of a non-developable ruled surface is tangent to the surface at just one point of 
the generator. For, the plane can be regarded as determined by the points y, 
z, and some point 2'+%' because the points y, z, y', z' are not coplanar; the 
tangent plane at the point z+uy coincides with this plane if, and only if, 
u = k. We see, then, that the tangent planes of a non-developable ruled surface 
may be assembled into a single infinity of axial pencils, whose axes are the 
generators of the surface. 

The cross ratio of four points P x given by equation (14) with a fixed value 
of t and four distinct values of u is equal to the cross ratio of these values of 
u, which is also equal to the cross ratio of the corresponding four points 
z'-f uy'. If the ruled surface R is not developable, this cross ratio is further 
equal to the cross ratio of the tangent planes of the surface at the first 
four points. So we have the correlation* of Chasles: 

The cross ratio of four points on a generator of a non-developable ruled sur- 
face is equal to the cross ratio of the four planes tangent to the surface at these 
points. 

* Chasles, 1839. 1, p. 53. 



101 RULED SURFACES 41 

Any curve C x on a ruled surface (see Fig. 6) in a linear space S n , except 
the curve C y and the generators, can be defined by placing u = u(t) in equa- 
tion (14). The osculating linear space S k at a point P x of the curve C x is de- 
termined by the points re, re', . . . , x (k \ where 

x' = z'+uy'+u'y , 
i x"=z"+uy"+2u'y'+u"y, 
(16) 



The linear space of least dimensions that contains a set of linear spaces 
may be called briefly the ambient thereof. So tne ambient, or joining space, 
of two distinct points is the line joining them; and the ambient of a point 
and a line not in united position is the plane determined by them. 

We are now ready to state an important definition. The osculating* 
space S(k, r) with respect to an element E r of a curve at a point P x of a ruled 
surface R is the ambient of the osculating space S k at the point P x of every curve 
on the surface R that passes through P x and has at P x the same element E r (r<k). 
For example, the osculating space S(2, 1) with respect to an element E\ 
of a curve at a point P x of a ruled surface R is the ambient of the osculating 
plane at the point P x of every curve on the surface R through P x having at 
P x the same tangent line. Sometimes we say that the space (2, 1) is in the 
direction of the curve. In particular, when r = 0, our definition becomes 
the following statement. The osculating space S(k, 0) at a point P x of a ruled 
surface R is the umbient of the osculating space Sn at the point P x of every curve 
on the surface R through P x . For example, the osculating, or tangent, space 
S(l, 0) at a point P x of a ruled surface R is the ambient of the tangent line 
at the point P x of every curve on the surface R through P x ; so the space 
S(l, 0) is the tangent plane at the point P x , of the surface R. It may be re- 
marked that in the definition of the space S(k, r) the ruled surface could be 
replaced by a general surface or variety (see 52, Chap. VII). 

It is a fundamental prpblem to determine the dimensionality of the 
space S(k, r) just defined. Every curve on a ruled surface R, through a 
point P x of R and having at P x the same element E r , has at P x the same 
values of u, u', . . . , u (r) ; but two such curves usually have different 
values of u (r+l \ . . . , u (k \ An upper limit on the dimensionality of the space 
S(k, r), which is ordinarily the actual dimensionality, can be determined 
by holding u, u' y . . . , u (r) fixed while u (r+1 \ . . . , u (k ^ vary in equations 

* Bompiani, 1914. 2, p. 307. 



42 PROJECTIVE DIFFERENTIAL GEOMETRY 

(16), and thus determining the greatest number of linearly independent 
points in the space S(k, r). In this process x, x', . . . , z (r) are fixed. But 
w (r+1) appears in the formula for x (r+1) as the coefficient of y, so that the point 
y can be taken as one of the points sought; in the presence of the point y, the 
point x can be replaced by the point z. Similarly, u (r+l \ u< r+2) appear in the 
formula for z (r + 2) as the coefficients of y', y respectively, so that the point y' 
can be taken as one of the points sought; in the presence of the points, y, y' 
the point x 1 can be replaced by the point z'. Continuing this reasoning we 
come to the last stage where we see that u (r+1 \ . . . , u^ appear in the for- 
mula for z (fc) as the coefficients of y( k ~ r ~ l \ . . . ,y', y respectively, so that 
the point y (k ~ r -V can be taken as one of the points sought; in the presence 
of the points ?/, y', . . . , y( k ~ r -V the point z ( *~ r ~ 1) can be replaced by the 
point z (k ~ r ~ l \ Then, in the presence of y, z and their first k rl deriva- 
tives, the point x (fc ~ r ^ can be replaced by the point z (k ~ r) -\-uy (k ~ r) ; the point 

x ( k -r+l) by the pomt z (*-r+l) + ^(fc-r+l) + (fc_ r+1 ) u y*-^ and go Qn until 

finally the point x w is replaced by the last of the points mentioned in the 
following statement. 

The osculating space S(k, r) with respect to an element E r of a curve at a 
point P x of a ruled surface R in a space S n (n^2kr) is determined by the 
2k r+1 points 



+ (k-r+2)u f y< k - r + + (k-r+2)(k-r+l)u"y^/2, 



and is therefore ordinarily a space S 2k - r . 

In particular, the space S(k, 0) is ordinarily the space Szk of even dimensions 
which is determined by the 2k-}- 1 points 

Then the space S(2, 0) is the space 4 of the points y, y', z, z', z" -\-uij"; 
as we have already seen, the space S(l, 0) is the tangent plane, which is 
determined by the points y t z, z'-\-uy'. 

A certain osculating linear space along a generator, rather than at a point, 
of a ruled surface will now be defined. The ambient of the spaces S(k, 0) at 
all points of a generator I of a ruled surface R is studied by allowing u to 



11] RULED SURFACES 43 

vary in formulas (18). This ambient is thus seen to be determined by the 
2k+2 points 

(19) y, y', . . . , 2/ (A) , z, *',... *<*> , 

and is therefore ordinarily a space Szk+i- This space of odd dimensions is 
called the osculating space $2/0+1 along the generator I of the ruled surface R. 
This is in fact the linear space of least dimensions that contains the osculating 
spaces Sk at points of I of all curves on R that intersect I. In particular, the 
osculating space S 6 along a generator I of a ruled surface R is seen to be the 
ambient of the osculating planes at points of I of all curves on R that inter- 
sect I. Finally, the osculating, or tangent, ordinary space 83 along a genera- 
tor / of a ruled surface R is the ambient of the tangent planes of R at all the 
points of Z, and is also the ambient of the tangent lines at points of the 
generator I of all curves on the surface R that intersect I. Sometimes this 
space 3 is thought of as determined by the generator I and a consecutive 
generator. 

11. The differential equations of a ruled surface in 83. Wilczynski's 
system of differential equations, defining a ruled surface in ordinary space 
S s except for a projective transformation, is fundamental in this and the 
next three sections. In this section canonical forms are obtained for the 
differential equations by means of certain transformations of the inde- 
pendent and dependent variables which do not disturb the surface. 

Let us consider* a non-developable ruled surface R with parametric 
vector equation (14), immersed in ordinary space $ 3 . The determinant 
(y, z, y f , z') does not vanish identically, because y, z, satisfy no relation of 
the form (15J. Consequently it can be shown that y, z satisfy a system of 
differential equations of the form 



(2o) 



in which the coefficients are scalar functions of the parameter t, and accents 
indicate differentiation with respect to t. In fact, if the four pairs of co- 
ordinates 2/ t , Zi (i = l, . . . , 4) are substituted in turn in the first of equa- 
tions (20) the resulting four linear algebraic equations can be solved unique- 
ly for the coefficients pu, Pit, <?n, #12; and the other coefficients can be de- 
termined similarly. Then R is called an integral ruled surface of system (20), 
and every projective transform of R is also an integral ruled surface of (20). 
Conversely, when equations (20) are given, the theory of differential equa- 
* Wilczynski, 1906. 1, Chaps. IV, V, VI. 



44 PROJECTIVE DIFFERENTIAL GEOMETRY 

tions tells us that these equations possess four pairs of solutions 2/, z\ 
(i=l, . . . ,4) forming a fundamental set. When these solutions are inter- 
preted as the coordinates of two points P y , P zy we see that equations (20) 
define a ruled surface in space S s except for a projective transformation. 
Let us consider the transformation of dependent variables, 

(21) 2/ = ar,+/?r, Z = W+K (A = a5-/?7^0) , 

in which the coefficients a, 0, 7, 6 are scalar functions of t. The effect of this 
transformation on equations (20) is to produce the system of equations 



^ 



Geometrically, the transformation (21) is a change of director curves on an 
integral ruled surface from C y , C z to (?, CV , leaving the surface unchanged. 
Therefore systems (20) and (22) have the same integral ruled surfaces. 

The transformation (21) can be used to simplify the fundamental differ- 
ential equations. If a, 7 and 0, 5 are two pairs of solutions of the equations 



(23) 

for p, a, then equations (22) become 

(24) { 

wherein the functions u ik are defined by the formulas 



If we were to solve equations (24) for r)", ", we would obtain two equations 
of the same form as (20) but with the terms in V> f ' missing. Therefore we 
have the theorem: 

It is possible by means of the transformation (21) to reduce system (20) to a 
canonical form for which p lk = (i, k = 1, 2) . 

We next deduce a second canonical form of system (20). If equations 



11] RULED SURFACES 45 

(22) are solved for if', " the result is another system of the same form as 
(20) whose coefficients 71-^, Xik are given by the equations 



(26) 



and the equations obtainable therefrom by the substitution 
(y 1 77 a ft a p\ 



(27) V 2 f 7 b er, 

in which a, 6, p, a will be defined later in equations (36) and (41) ; this substi- 
tution leaves all the systems of equations in this section invariant. By 
means of these results it can be shown, although we shall omit the calcula- 
tions, that the effect of the transformation (21) on the functions u lk defined by 
the formulas (25) is given by 



(28) 



wherein the dash indicates the transformed functions. If the equation 

(29) Wi 2 X 2 + (wu - w 22 )X - W2i = 

has distinct roots so that its discriminant 64, denned by the formula 

(30) Bi = (MH - W2 



is not zero, let us take one of these roots for y/a and the other for 5//9. Then 
we have ^12=^21=0. So we reach the conclusion: 

It is possible by means of the transformation (21) to reduce system (20) to a 
canonical form for which u\z = u^\ = 0, provided O^Q. 

The most general transformation (21) preserving the canonical form of 
system (20) characterized by the conditions Ui 2 = W2i = is subject to the 



46 PROJECTIVE DIFFERENTIAL GEOMETRY 

restrictions f}5 = ya = Q when Un uzz^O. It is possible to make any trans- 
formation (21) for which = 7 = 0, a5^0 without disturbing this canonical 
form. When the differential equations are in this canonical form, the 
director curves are definitely chosen, except for a mere interchange; their 
geometrical description will be given in Section 13. 
Let us next consider the transformation of parameter, 

(31) r = r(t) (TVO) . 

The effect of this transformation on system (20) is to produce another sys- 
tem of the same form whose coefficients, indicated by dashes, are given by 
the formulas 



, 

= l, 2; 

The effect of the transformation (31) on the functions Ui k is given by 

(33) T' 2 tf tt = w tt +2a/-co 2 , T' 2 u lk = u lk . 

The function 6*, which can be shown by equations (28) to be absolutely 
invariant under the transformation (21), is transformed by (31) into 4 
according to the formula 

(34) r' 4 04 = 4 , 

and is therefore a relative invariant under the total transformation (21), 
(31). The geometrical significance of the absolutely invariant equation 
04 = will be pointed out in Section 13. 

It is possible to simplify still further the canonical form of system (20) 
characterized by the conditions ^12 = ^21 = 0. The transformation 

(35) 2/ = ar7, Z = 5f, T = r(0 (afirVO) 

leaves these conditions invariant and changes the coefficients pn, p 22 into 
Pn, 7>22 according to the following formulas, which may be written by the 
aid of equations (26), (32), and the substitution (27): 



Therefore, if a, 5 are solutions of the equations 
2a'-t-(pii+o>)a = , 



12] RULED SURFACES 47 

with T and hence co still arbitrary, we have pn=p 2 2 = and still have at our 
disposal the transformation 

(36) r = r(t) , y = arj/(r / )^ , * = &?/(/) 1/2 (a, 6 = consts.) . 



By means of this transformation we can make 4 = 1 by choosing T so that 
r' 4 = 4 . Or instead we can make u\\ 22 =1 by choosing T so that r' 2 = 
Un u 2 2. Or as a third possibility we can make fe4-W22 = by choosing r 
so that 



12. The asymptotic curves and the osculating quadric. The asymptotic 
curves are of primary importance in the projective differential geometry 
of a ruled surface in ordinary space. Study of their properties leads, in this 
section, to the definition of a certain regulus called the asymptotic regulus 
along a generator of a ruled surface. There is a second regulus called the 
osculating regulus also associated with the* generator. These two reguli lie 
upon the same quadric surface, called the osculating quadric along the 
generator of the surface. When the generator is allowed to vary over the 
surface we arrive at the definitions of theflecnode congruence and of a deriva- 
tive ruled surface. 

In ordinary space $3 let us consider a non-developable ruled surface R 
which has the parametric vector equation (14), and which is an integral 
surface of system (20). Using the expression for x given in equation (14), 
let us replace v by in equations (9), (10), and calculate the determinants 
L, M, N, defined by (10). Then (9) shows that the following statement is 
true : , 

The asymptotic curves on the ruled surface R consist of the generators and 
the oo i curves defined by the differential equation 

(37) 2w' = p 21 +(pii-p22)u-pi2U 2 (u f = du/dt) . 



The curves (37) are called the curved asymptotics, or sometimes simply the 
asymptotics, on the surface R. The director curves C v> C z are asymptotic 
curves in case pi 2 = 21 = 0; and every curve u const, is an asymptotic curve 
in case 

P12 = P11-P22 = P21 = . 

Since (37) is an equation* of Riccati, the cross ratio of four particular solu- 
tions is constant, and the theorem j of P. Serret follows: 

* Goursat-Hedrick, 1917. 1, p. 12. 
t Serret, 1860. 1, p. 169. 



48 PROJECTIVE DIFFERENTIAL GEOMETRY 

The cross ratio of the four points in which four curved asymptotics on a ruled 
surface in ordinary space intersect a generator is the same for all generators of 
the surface. 

The locus of the asymptotic tangents at the points of a generator I of a 
ruled surface R will now be proved to be a regulus, which will be called the 
asymptotic regulus along the generator I of the surface R. It is sufficient to 
prove that these tangents lie on a quadric surface. Any curve C x , except C y 
and the generators (see Fig. 6), on the surface R can be defined by placing 
u = u(t) in equation (14). The tangent line of the curve C x , at the point P x 
where C x crosses the generator I, is determined by P x and the point X de- 
fined by placing 

X=x'+vx (v scalar) . 

If the curve C x is an asymptotic curve we find by means of equations (14), 
(37) the expression 



wherein the coefficients xi, . . . , x* are given by 
(38) \ Xl UV P21 Pn P " U 

{Xz = V , Xs~U , 4=1 . 

These coefficients are the local coordinates of the point X referred to the 
local tetrahedron y, z, y', z' with suitably chosen unit point. When u } v 
vary, t being fixed, equations (38) are the parametric equations of a quadric 
surface whose algebraic equation is found, by homogeneous elimination of 
u } v, to be 

This completes the proof. 

We now state the definition of the osculating regulus. The osculating 
regulus along a generator I of a ruled surface R is the limit of the regulus de- 
termined by I and two neighboring generators of R as each of these independently 
approaches I, remaining on R. The quadric (39) is called the osculating 
quadric along the generator I of the ruled surface R because the osculating 
regulus lies upon it, as we shall now demonstrate. Referring to Figure 7, 
let us consider a point Y near the point P v on the director curve C v of the 
ruled surface R } and consider also the corresponding point Z near P z on C z . 
Taylor's expansion gives 



1 12] 



RULED SURFACES 



49 



where A is the increment of t corresponding to displacement on the curve 
C v from the point P v to the point Y. Equations (20) express y", z" as linear 




FIG. 7 



combinations of y, z, y', z'\ after differentiating equations (20), the third 
derivatives y'", z'" can be so expressed; and so on for derivatives of y, z of 
any required order. The point defined by the formula 



hY+kZ 
is on the generator YZ of the surface R, and we find 



(h, k scalars) 



are given by 



where the coefficients x\, . 



(40) 



Demanding that the general equation of a quadric surface be satisfied by 
the power series (40) for xi, . . . , x\ identically in h, k and A as far as the 
terms in A 2 , we obtain equation (39) again, thus completing the demonstra- 
tion. We observe that an ordinary tangent of a ruled surface intersects two 
consecutive generators, and that an asymptotic tangent intersects three consecu- 
tive generators at its point of contact. 



50 PROJECTIVE DIFFERENTIAL GEOMETRY 

It is sometimes convenient to use the local tetrahedron (see Fig. 7) y, z, 
p, a, the points p, a being denned by placing 

(41) p = 2y'+pny+pi2Z , <T = 2z'-fp 2 i2/-|-p22Z 

If a point has coordinates xi, . . . , z 4 referred to the tetrahedron y, z, y f , z', 
and coordinates 2/1, . . . , 2/4 referred to the tetrahedron y } z, p, <r, the identity 



where p is a proportionality factor, gives the equations of transformation 
between the tetrahedrons, namely, 

f pZl = 



We may sometimes neglect to write the factor of proportionality when its 
presence, as in these formulas, is not essential. 

Referred to the tetrahedron y, z, p, a the equation of the osculating 
quadric becomes 

(43) 



This equation can be used to show that the simple skew quadrilateral ypcrz 
lies entirely on the osculating quadric. That the lines yp and za do so is evi- 
dent when one observes that the local equations of these lines are ?/ 2 = 2/4 
and 2/1 = 2/3 = respectively. The line yz or 2/3 = 2/4 = of course lies on the 
quadric, and so also does the line pa or 2/1 = 2/2 = 0. It follows that the lines 
yp and za are asymptotic tangents, and the line pa is a generator of the osculating 
regulus. 

We propose to study the line pa a little more closely. The effect of the 
transformation (31) on p, a is found by the formulas (32) to be given by 



where p, <r are respectively the transforms of p, a. Let us consider any point 
p-\-hy on the line yp, and any point a-\-kz on the line za, and let us impose 
the condition that the line joining these two points be a generator of the 
osculating regulus. Substituting the local coordinates h, Xfc, 1, X of an 
arbitrary point on this line in equation (43) we find that the condition to be 
imposed is h = k. We now observe that if u = h, the transformed line P 
coincides with this arbitrary generator of the osculating regulus. We thus 
have proved the theorem : 



13] RULED SURFACES 51 

The line pa can be made to coincide with any prescribed generator of the 
osculating regulus by choosing the independent variable suitably. 

Let the generator yz of the ruled surface R now vary over R. There is 
clearly a one-parameter family of osculating reguli, and the totality of their 
generators is a two-parameter family, or congruence, of lines called the 
flecnode congruence of the surface R (see Exs. 6, 7, 8). With a suitable defi- 
nite choice of the independent variable, the locus of the line per is an arbi- 
trary ruled surface of this congruence, which is called a derivative ruled 
surface of the original ruled surface R, and is sometimes called the deriva- 
tive ruled surface of R with respect to the independent variable employed. 
We remark incidentally that the totality of the asymptotic tangents of the 
surface R is also a congruence, called the asymptotic congruence of R. 

13. The flecnode curves and the flecnode transformation. Continuing 
the geometrical investigation of the preceding section, we shall occasionally 
find it convenient to denote a ruled surface with director curves C v , C z by 
R yz . Among the asymptotic tangents at the points of a generator l yt of the 
surface R yz there are two that are distinguished from the rest by the prop- 
erty of intersecting more consecutive generators than is usual for an asymp- 
totic tangent. The points of contact of these two tangents are called flec- 
node points, or sometimes simply flecnodes, because at such a point the 
tangent plane of the surface R yz intersects R yz in a curve having at the 
point an inflexion and a node (see 17, Chap. III). In the present section 
we shall study the locus of these flecnode points, called the flecnode curves, 
and on the basis of the flecnodal properties of a ruled surface shall define a 
transformation of ruled surfaces called the flecnode transformation. 

The definition of a flecnode point may be concisely stated as follows. A 
flecnode point on a ruled surface in ordinary space is a point at which the 
asymptotic tangent intersects four consecutive generators. Since four skew 
lines in ordinary space have two straight line intersectors,* it follows that 
there are two flecnode points on each generator of a ruled surface in ordinary 
space. These two flecnode points may, of course, be real and distinct, real 
and coincident, or imaginary. Demanding that equation (39) be satisfied 
by the power series (40) for x\, . . . , z 4 conditionally in h, k and identically 
in A as far as the terms in A2 3 we obtain, using the formulas (25), 

(44) UiJP (un Uz2)hk u 2 ik 2 = . 

If h/k is a root of this equation, then the point hy+kz is a flecnode point on 
the generator l yz . Multiplying together the two linear expressions in y } z with 

* Snyder and Sisam, 1914. 1, p. 143. 



52 PROJECTIVE DIFFERENTIAL GEOMETRY 

irrational coefficients thus obtained for the flecnodes, we find for the ana- 
lytic representation of the flecnodes a quadratic expression with rational 
coefficients. This expression is a co variant, i.e., is invariant under the total 
transformation (21), (31), as is evident from its geometrical definition, and 
can be proved analytically. Thus we obtain the following theorem : 

Theflecnode points on a generator l vz of a ruled surface R yz in ordinary space 
are given by the factors of the covariant 

(45) Wi2Z 2 + (MII- u^yz -u^y* . 

The flecnode points are coincident in case 4 = 0. Thus we obtain the geo- 
metrical significance of this equation promised in Section 11. We shall ex- 
clude this case and suppose hereinafter, unless the contrary is stated, that 
the flecnodes are distinct, so that 04/^0. 

The locus of the flecnode points on a ruled surface is a curve which ordi- 
narily intersects each generator twice. Since in the neighborhood of such a 
generator there are two distinct parts of the locus, we shall speak of the 
locus as the flecnode curves. These are the director curves C v , C z in case Wi 2 = 
w 2 i = 0. It follows, moreover, from the definition of a flecnode point that 
every straight line intersecting < 1 generators of a ruled surface in space 83 
is a flecnode curve. Therefore every point on a generator of a quadric surface 
is a flecnode point. Consequently, if the surface R yz is a quadric, the co- 
variant (45) is indeterminate, so that for a quadric we have the necessary 
conditions ^12 = ^21 = ^11^22 = 0, which may be shown by the reader to be 
also sufficient by showing that when they are satisfied the curved asymp- 
totics (37) are straight lines. 

At a flecnode point the tangent of the flecnode curve is an ordinary two-point 
tangent whose locus is a developable surface with the flecnode curve as edge 
of regression, whereas the asymptotic tangent is a four-point tangent, called 
the flecnode tangent, whose locus is ordinarily a non-developable ruled sur- 
face called a flecnode surface. In order to find the parametric vector equa- 
tions of the flecnode surfaces of a ruled surface R yz , let us suppose that 
system (20) is in the canonical form for which 

C Ul 2 = UK = Pn = p 2 2 = , Pi2 = 2^2 , $1 = 2^21 , 

(46) < 



Now the director curves C v , C z are the flecnode curves, as in Figure 8. At a 
point P y the tangent of the flecnode curve is determined by the point y f , 



113] 



RULED SURFACES 



53 



and the flecnode tangent by the point p. The parametric vector equations 
of the flecnode surfaces are therefore, respectively, 



(47) 



the points p, $ being the generating points, and v, w being scalar parameters. 
If p\2 0, the flecnode curve C v reduces to a straight line, since the first 



of equations (20) becomes 



In this line coincide the tangent 




FIG. 8 



yy' of the flecnode curve and the flecnode tangent 2/p, which in this case is 
fixed and does not generate a proper surface. We shall exclude this case and 
the similar case involving the line zv, by supposing pi2?>2i 5^0. 

The differential equations of the form (20) for the flecnode surface R yp gener- 
ated by the line yp are found by use of equations (20), (46) to be 



(48) 



where the coefficients denoted by capital letters are given by 



1 = 25n- 012521 - 



For this system of equations the functions U*k t analogous to the functions 
u lk defined by the formulas (25), can be calculated. We find 



(49) 



54 PROJECTIVE DIFFERENTIAL GEOMETRY 

The flecnodes on the line yp are found, by writing formula (45) with U tk in 
place of u lk and with p in place of z, to be P y and the point (Un t/ 22 )p 
Uziy. Consequently C y is aflecnode curve on the ruled surface R vp as well as on 
R yz . The line yz can be shown to be the flecnode tangent of the surface 
R yp at the point P v by calculating the function P for system (48) analogous 
to the function p defined by the first of equations (41) for system (20). The 
result is P= ZQizypuz. By means of the substitution (27) it is easy to 
write the equations for the flecnode surface R Zff , and to prove that C z is a 
flecnode curve on R za as well as on R yz . 

We are now in position to define the flecnode transformation of ruled 
surfaces in ordinary space. The original surface R yt has for its flecnode 
surfaces R yp and R Zff . Moreover, R yp has for its flecnode surfaces R yz and 
another, which has for its flecnode surfaces R yp and another, and so on. 
Thus we obtain from the ruled surface R yz a sequence of ruled surfaces 
each of which has the two adjacent surfaces as flecnode surfaces. Such a 
sequence is called aflecnode sequence, and two surfaces in it are said to corre- 
spond by the flecnode transformation. 

14. The osculating linear complex of a ruled surface in 83. We first state 
the following definition. The osculating linear complex along a generator l yz 
of a ruled surface R yz in ordinary space 83 is the limit of the linear complex 
determined by l ye and four neighboring generators of R yz as each of these inde- 
pendently approaches l yz , remaining on R yz . The local equation of the oscu- 
lating linear complex will now be obtained in the line coordinates defined in 
Section 7; and a few consequences will be deduced. 

Let us suppose that equations (20) are in the canonical form for which 
equations (46) are valid. Then differentiation of equations (20) and sub- 
stitution from equations (41) enable us to express derivatives of y and z of 
any required order as linear combinations of y, z, p, a. Actual calculation 
gives 



where the coefficients L, A, B, C, will not be needed, and M, D are defined 
by 

M= -q( 



14] 



RULED SURFACES 



55 



As in Section 12 let us write Taylor's series for the coordinates of a point Y 
near the point P v on the director curve C y . Then substitution of the expres- 
sions just found for the derivatives of y enables us to write Y in the form 



where the local coordinates yi, 
expansions 



of the point Y are given by the 



(50) 



1/4= - 



Similar expansions for the coordinates Zi, . . . , 24 of a point Z near P z on C t , 
referred to the local tetrahedron ?/, z, p, o-, can be written easily by means of 
the substitution (27) augmented by interchanging the subscripts 3 and 4. 
Then for the coordinates u lk of the line YZ, denned by 



(i, 



l, . . . , 4) 



we find the expansions 



(51) 



Writing the general equation of a linear complex and demanding that it be 
satisfied by the power series (51) for u lk identically in A2 as far as the terms 
in A 4 , we find the equation of the osculating linear complex, 



(52) 



The introduction of the osculating linear complex paves the way for 
further extensions of the theory of ruled surfaces. The null system of the 
osculating linear complex (52) has the equations 



(53) 



2 = ~ 



4 = 



56 PROJECTIVE DIFFERENTIAL GEOMETRY 

At any point (h, k, 0, 0) on the generator l yz the equations of the tangent 
plane and the null plane are respectively 



These planes coincide in case p^h* pnk 2 = Q. The two points puz^ 
thus determined are called the complex points of the generator l v ,. At each 
of them the tangent plane of the surface corresponds to its point of contact 
in the null system of the osculating linear complex. The locus of the com- 
plex points is known as the complex curves on the surface. The two points 
Pi22 2 -fp2i2/ 2 separate harmonically the complex points and also separate 
harmonically the flecnodes y, z. On this account they are called the involute 
points of the generator l yz , and their locus is defined to be the involute curves 
on the surface. 

The reader who wishes to refer to a fuller treatment of ruled surfaces in 
ordinary space than that which has been found here should consult* Wil- 
czynski's book. Attention is especially directed to Chapter VIII on the 
conditions that a ruled surface may belong to a linear complex, or to a 
linear congruence, etc. Moreover, the reader who wishes to see a somewhat 
different method of handling the subject of ruled surfaces in ordinary space, 
namely, the method of differential forms, should consult f Chapter IV of the 
first volume of the treatise by Fubini and tech. There are to be found in 
that chapter further geometrical results of great interest. 

16. Quasi-asymptotic curves. An unspecialized ruled surface in a linear 
space of n dimensions S n (n>3) does not have on it an asymptotic curve 
other than the generators. The important role of the asymptotic curves on 
a ruled surface in space $ 3 suggests the desirability of finding a substitute 
for them in the theory of ruled surfaces in hyperspace. Bompiani has de- 
fined on a ruled surface in space S n certain curves, called quasi-asymptotic 
curves, with properties analogous to those of asymptotic curves. These 
quasi-asymptotic curves are the asymptotic curves when n 3, and so are 
actually a generalization of asymptotic curves. They will be discussed in 
this section. 

At a point of a curve on a surface in space S n the osculating plane of the 
curve and the tangent plane of the surface intersect in the tangent line of 
the curve, and therefore the ambient of these two planes is ordinarily a 
space $ 3 ; but for an asymptotic curve the ambient is a plane, because the 
two planes coincide. This property of asymptotic curves suggests the fol- 
lowing considerations which lead to the definition} of quasi-asymptotic 

*Wilczynski, 1906. 1. 

t Fubini and Cech, 1926. 1. J Bompiani, 1924. 2, p. 313. 



15] RULED SURFACES 57 

curves. At a point x on a ruled surface R yz with parametric vector equa- 
tion (14) in space S n let us consider the space S(k, 0), which is ordinarily 
a space Szk determined by the 2fc-fl points (18); and let us also consider 
the osculating space S n -k of a curve C X) which is determined by the n fc-f 1 
points x, x'j . . . , x (n ~ k \ Wishing to discover the dimensionality of the 
ambient of these two spaces, we observe that the first k+l of the latter set 
of points are linearly dependent on the points (18). Therefore the ambient 
of the space S(ti, 0) and the osculating space S n - k is ordinarily the space S n . 
A quasi-asymptotic curve A k , n - k on a ruled surface is defined to be a curve such 
that at each of its points the ambient of the space S(k, 0) of the surface and the 
osculating space S n - k of the curve is a hyperplane S n -i. It is clear that this 
definition is valid only when fc>0; moreover, when the space S(k, 0) is a 
space Szk one must also have k<n/2. 

If the curve C x is a quasi-asymptotic curve then the n+ 1 points in the 
set composed of the points (18) and x^ k+l \ . . . , x (n ~ k ^ lie in a hyperplane 
and are therefore linearly dependent. Consequently the determinant of 
order n-\-l made with their coordinates must vanish. Expressing this fact 
by writing only a typical row of the determinant within parentheses, we 
obtain the differential equation of the quasi-asymptotics Ak, n -k in the form 

(y, y f , - 

, . 

( } 



Since this differential equation for u as a function of t is of order n 2k, it 
follows that there are < n ~ 2k quasi-asymptotics A ktn - k on a ruled surface in 
space S n . One of these can be determined uniquely by assigning for it n 2k 
points suitably chosen on the surface ; or else by assigning an element E n - 2 k-i 
at one point on the surface; or else in still other ways. 

Placing n = 3 and fc = l, we observe that the curves Ai )2 are the curved 
asymptotics that exist on a ruled surface in ordinary space. More generally, 
if n is odd, let n = 2m 1. If we now place fc = m 1, then nk = m and 
there are observed to be oo 1 curves A m -i tm on a ruled surface in space Stm-i', 
their differential equation is 

7l( 2) y y f (m 2) 

(55) < ..z 



This equation for u as a function of t is of the first order; therefore one 
curve A m -i >m passes through each point of the surface. Moreover, this 



58 PROJECTIVE DIFFERENTIAL GEOMETRY 

equation is an equation of Riccati, so that we have an extension of the 
theorem of Serret: 

The cross ratio of the four points in which four quasi-asymptotics A m -i, m 
on a ruled surface in space S 2m -i intersect a generator is the same for all genera- 
tors of the surface. 

If n is even, let n = 2m. If we now place k = m, then nk = m and there is 
observed to be a unique curve A m , m on a ruled surface in space Sz m ; this curve 
is the locus of the point x given by equation (14) when u satisfies the equa- 
tion 

(56) (y, y',..., y<"-, z, *',..., ~'-, z^+uy^) =0 . 

At points of this curve the space S(k, 0) of the surface, which is determined 
by the points (18), is a space Su-i instead of the usual space $2*, since the 
points (18) are precisely the points which equation (56) asserts to be linearly 
dependent. In particular, if n = 2 and k = l, then w=l and the unique 
curve ^4.i ( i on oo 1 lines in a plane is their envelope. Finally, if n = 4, fc = 2, 
then m = 2 and the unique curve A 2 , 2 on a ruled surface in a space $ 4 is the 
locus of the point (14) when u satisfies the equation 

(57) (y,y',z,z' ) z"+uy")=Q. 



EXERCISES 

1. In ordinary space a developable and a curve are dual configurations. Com- 
pare Sections 6 and 9 from this point of view. 

2. The equation of the developable of which the twisted cubic (I, 26) is the edge 
of regression is 



This developable is of class 3 and order 4, the class of a developable being the num- 
ber of its tangent planes through any point, and the order being the number of 
points in which a straight line meets it. 

3. The cubic ruled surface x\xz ^4 = has two distinct rectilinear directrices 
y 1 = 0, 2/2 = 0, 1/3 = t 2 , 2/4=1 and zi=l, z 2 = t, 23 = 0, 24 = 0. The differential equations 
of the form (20) for this surface are y" y'/t = 0, z" = 0. The asymptotic curves on 
the surface are unicursal quartics each of which crosses each generator in two points 
separating harmonically the flecnodes on the generator. 

WILCZYNSKI, 1906. 1, p. 145 

4. The cubic ruled surface x\(x\x^-\-x 2 x^+x\=Q has one rectilinear directrix 
2/i = 0, 2/2 = 0, 2/3 = 2, 2/4 = 1 an d a director curve z\ 1, z 2 = t, 23 = 0, z 4 = 2 2 . The 



RULED SURFACES 59 

differential equations of the form (20) for this surface are 2/" = 0, z"-\- 2ty f 2y-Q. 
The asymptotic curves are twisted cubics, while the flecnode curves coincide in the 
rectilinear directrix. WILCZYNSKI, 1906. 1, p. 145 

5. Considering an integral surface of system (20) with pik 0, show that the local 
coordinates of a point near P x on the curve C x defined by equation (14) with u = u(t), 
referred to the tetrahedron y, z, y', z', are given by 



The equation of the tangent plane of the surface at P x is x^ wz 4 = 0; the equations 
of the tangent line of C x are 3 ux\=Xi ux 2 u'x\=0\ and the equation of the 
osculating plane of C x is 

2u'(xi ux 2 u'x^) (z 3 uxtW+qizU* (qn q z2 )u g 2 i] = . 

Every asymptotic curve u= const, has three coincident points in the tangent plane, 
and two in its tangent line. At a flecnode the asymptotic curve has four coincident 
points in the tangent plane and three in its tangent line, its osculating plane being 
indeterminate. A flecnode is therefore an inflexion point on the asymptotic curve 
through it. 

6. By means of equations (20), (39) with p t k = Q show that the points ?;, f defined 
by "n y'+hy, ^ z f -\-hz determine a general generator l^ of the osculating regulus 
along a generator l yz of the ruled surface R vz , so that the point <f> defined by ^= 77+ Af 
is a general p$int on l^ . If, as t varies, l^ describes a developable of the flecnode 
congruence of R yz , and if P v is the corresponding focal point of l^ , then <?' must be 
a linear combination of y and z only; hence we obtain two conditions: 



Eliminating hf h?, prove that l^ has two such foci which are on the flecnode tan- 
gents, so that the lines of the flecnode congruence are tangent to both of the flecnode 
surfaces, the points of contact of each line being its foci. (On this account the flec- 
node surfaces are called the focal surfaces of the flecnode congruence. Moreover, the 
flecnode congruence receives its name from its relation to the flecnode surfaces.) 
Eliminating X, prove that the lines of the flecnode congruence can be assembled 
into oo i developables in two ways. 

7. By means of equations (37), (47) show that the differential equation of the 
asymptotic curves on the integral ruled surface R yp of the equations (48) is 2t/ = 
2(#n-h<?22) 7>i2?>2i-M 2 ; show that the equation of the asymptotics on the surface 



60 PROTECTIVE DIFFERENTIAL GEOMETRY 

R gff is the same with v replaced by w. The asymptotics on R yp and R Zff therefore 
correspond in the sense that, if a point of contact of a generator of the flecnode con- 
gruence describes an asymptotic curve on one surface, the other point of contact 
does also on the other. (A congruence having this property is called a W congruence.) 

8. Using the canonical form of system (20) described by equations (46), and 
placing A = (q(\ q^)/ (qnqw), B = q i2 /pi 2 , show that the point p+Ay/2 on the 
flecnode tangent l yp at the point P y is the harmonic conjugate of P y with respect to 
the other flecnode p+(A 2B)y on l yp and the point p -\-2By which lies on the same 
generator of the osculating regulus z 2 z 3 ZiZ 4 = as the point <r+2Bz where the 
flecnode tangent l zff at the point P z meets the osculating plane Pi&^+pnxs 2q n xt = 
of the flecnode curve C v at P y . The point <r+Az/2 can be defined similarly on the 
line hff. The line joining these two points is a generator of the osculating regulus 
and generates a ruled surface of the flecnode congruence (called the principal ruled 
surface of the flecnode congruence) ; this is the derivative ruled surface R pa in case the 
independent variable is chosen so that 04 = const. 

9. In a linear space of four dimensions 84 three skew lines have just one straight 
line intersector. 

10. In space $4 the ruled surface defined, except for a projective transformation, 
by the differential equations 



z" = ay+bz+cy'+dz'+ey" , 

has on it an asymptotic curve if, and only if, 2e' = c-\-ed. The unique quasi-asymp- 
totic A 2, 2 on the surface is the locus of the point z-ey at which a line can be drawn 
intersecting three consecutive generators. The osculating plane at a point of the 
curve A 2)2 lies in the tangent space 83 of the surface along the generator through 
the point. This space 3 is the hyperplane employed in the definition of the curve 
A 2 , 2 . If there is an asymptotic curve on the surface this curve is the curve A 2|2 . 

11. From the equations of Exercise 10 obtain the equation 

z'" = Ay+Bz+Cy'+Dz'+Ey" , 

where 

A = a'+ad+ea , B = b'+bd+ep , C=c'+a+cd+ey , 

E=e'+c+ed+ee . 



Then show that the differential equation of the o> 2 quasi-asymptotics Ai t t on the 
ruled surface considered in that exercise can ordinarily be written in the form 

(u+e)[C+u y +3u"-u(D+ud)]-(c+2u'-ud)(E+u*+Zu')=Q . 



RULED SURFACES 61 

The osculating space 83 at a point of one of these curves contains the tangent plane 
of the surface at the point. A curve C x on the surface is a quasi-asymptotic A\ t $ 
in case at each of its points the space (2, 1) of the surface in the direction of the 
curve has four-point contact with the curve. 

12. In space 84 the tangent space $3 along a variable generator of a ruled surface 
osculates a curve (called the associated curve of the surface). If the quasi-asymptotic 
A 2 , 2 is chosen for the curve C z , so that e = 0, the associated curve is generated by 
the point X defined by 



in the notation of Exercise 10. If the curve A2.2 is not an asymptotic curve, the 
tangent line at a point of the associated curve intersects the corresponding generator 
yz at the point z, where the generator crosses A 2>2 ; and the osculating plane of the 
associated curve contains the generator. If the curve ^2,2 is an asymptotic curve 
but not a straight line, then the associated curve coincides with A 2,2- 

BOMPIANI, 1914. 2, p. 309 

13. At a point of a flecnode curve of a ruled surface R in space $3, consider the 
harmonic conjugate line I of the tangent of the flecnode curve with respect to the 
generator of the surface and the flecnode tangent. When the generator varies over 
the surface R, the line I generates a developable surface. Determine the edges of re- 
gression of the two developables thus associated with the surface R. 

WILCZYNSKI, 1906. 1, p. 233 

14. There exists a one-parameter family of ruled surfaces having one flecnode 
curve in common with a ruled surface in space 3. The surfaces of this family can 
be paired as in an involution, the double surfaces being the developable of tangents 
of the flecnode curve and the developable associated with this curve as described in 
Exercise 13. At each point of the flecnode curve the generators of the ruled surfaces 
are in involution. WILCZYNSKI, 1906. 1, p. 233 

15. A ruled surface in space $5, which has on it two curves C v , C z such that their 
osculating planes at points on each generator do not intersect, is an integral surface 
of a pair of equations of the form 



Study the surface by means of these equations. STOUFFER, 1913. 3, p. 204 

16. The two osculating planes of the flecnode curves, the two osculating planes 
of the complex curves, and the two osculating planes of the involute curves, at 
points of a generator of a ruled surface in space S 3 , belong to a pencil if, and only if, 
in the notation of equations (46), 



CARPENTER, 1928. 7, p. 481 



62 PROJECTIVE DIFFERENTIAL GEOMETRY 

17. The line of intersection of the two osculating planes of the flecnode curves at 
the points where these curves cross a generator I of a ruled surface R in space 83, 
the line that corresponds to this line in the null system of the osculating linear com- 
plex of R along /, and the generator I itself determine a quadric surface. This 
quadric intersects the osculating quadric of R along I in I and in a twisted cubic 
which passes through the complex points of I. The point corresponding to a general 
osculating plane of this cubic in the null system of the osculating linear complex of R 
along I generates another twisted cubic whose points are in one-to-one correspond- 
ence with the points of the first cubic. The lines joining corresponding points of 
these cubics form a ruled surface T, upon which the two cubics are asymptotic 
curves, and which belongs to a linear congruence, whose directrices cut each genera- 
tor of the surface T in two points separating harmonically the points in which the 
two cubics cross the generator. CARPENTER, 1923. 5, pp. 108 and 111 

18. As a generator I of a ruled surface R, not a quadric, in space $ 3 varies over 
the surface R, the envelope of the osculating quadric of R along the generator I 
consists of the surface R itself and of the two flecnode surfaces of R. 

19. Starting with the canonical form of system (20) for which ^ = 0, calculate 
the linear differential equation of the sixth order satisfied by the line coordinates 
of a generator of an integral ruled surface. Hence deduce conditions necessary and 
sufficient that an integral ruled surface of system (20) belong to a single non-special 
linear complex, to a single special linear complex, to a linear congruence with dis- 
tinct directrices, or to a linear congruence with coincident directrices. 

WILCZYNSKI, 1906. 1, Chap. VIII 

20. The most general transformation (21) that preserves the canonical form of 
system (20) characterized by the conditions p t /b = (i, k=l, 2) has a, /3, y, 5 all 
constants. 

21. Consider the hyperplane S n -i associated with each point of a quasi-asymp- 
totic curve Ak, n -k on a ruled surface in space S n . All these hyperplanes associated 
with the points of a curve Ak, n -k are the osculating hyperplanes of a curve 
Bk, n -k associated with Ak, n -k. Discuss the relations of these two curves, show- 
ing that each osculating space $ n -i- of the curve Bk, n -k contains the correspond- 
ing osculating space S n -k-i of the curve Ak, n -k (i=0, . . . , n k). 

BOMPIANI, 1914. 2, p. 315 

22. If a developable surface in space S n has on it a plane curve (not the edge of 
regression and not a generator), the developable is immersed in ordinary space. 
In general, if a developable surface in space S n has on it a curve (not the edge of re- 
gression and not a generator) immersed in a space Sk, then the developable is im- 
mersed in a space Sk+i (Q<k<n). 



RULED SURFACES 63 

23. (A ruled surface in space S n is said to have indices of developdbility h, k when 
the generators of the surface lie one by one in the osculating spaces Sh of a curve and 
each generator intersects the osculating space Sk which is in the osculating space 
Sh containing it.) For an ordinary ruled surface in space $3 the indices are 2, 1. 
For a developable surface the indices are 2, 0. For an ordinary ruled surface in 
space $4 the indices (see Ex. 12) are 2, 1 ; and if a ruled surface in space 84 has on 
it an asymptotic curve not a straight line, the indices are 2, 0. The maximum 
values of the indices are w/2, (n 1)/2 if n is even, and are (n+l)/2, ft 2 if n is odd. 

BOMPIANI, 1914. 2, p. 312 



CHAPTER III 
SURFACES IN ORDINARY SPACE 

Introduction. There are two well-recognized methods that have been 
used in developing extensive analytic theories of projective differential 
geometry. Since these methods are perhaps best exemplified by the theories 
of analytic non-ruled surfaces in ordinary space, it seems appropriate to 
make a few comments on them here. These methods may be designated as 
the American method, which is due to Wilczynski, and the Italian method, 
which is due to Fubini. 

The method of Wilczynski originated, as has already been indicated, in 
connection with curves and ruled surfaces in the years immediately pre- 
ceding 1906, the earliest memoir of interest in this connection appearing in 
1901. The method was extended about 1907 so as to be available for the 
study of so-called curved surfaces in ordinary space. In its most general form 
this method can be described as follows. When a configuration is to be 
studied, one starts with the parametric equations of the configuration, 
which express the projective homogeneous coordinates of a variable element 
of the configuration as functions of a certain number of parameters. The 
first step is to establish a completely integrable system of linear homogene- 
ous partial differential equations of which these coordinates constitute a 
fundamental set of solutions, the independent variables in the equations 
being the said parameters. These equations define the configuration except 
for a projective transformation, since the most general solution of the equa- 
tions can be expressed as a linear combination of the fundamental set of 
solutions with constant coefficients. The next step is to determine the most 
general group of transformations of the dependent and independent vari- 
ables in the equations that does not disturb the configuration, and to calcu- 
late the effect of this group of transformations on the differential equations. 
The third step is to find a complete and independent system of invariants 
and covariants; invariants are functions of the coefficients of the differential 
equations and of their derivatives, and covariants are functions of these and 
also of the dependent variables and their derivatives, that are unchanged 
by the group of transformations, except possibly for a factor depending only 
on the transformations. This work is purely analytic and may employ the 
Lie theory of continuous groups. Finally, the invariants and covariants 
are interpreted geometrically and used to study the geometry of the con- 
figuration in the manner explained in Section 2 in connection with curves. 

64 



SURFACES IN ORDINARY SPACE 65 

The method of Fubini may be said to have originated just before 1916. 
In fact, the method seems to have been introduced in a memoir published 
in 1914, in which Fubini proposed the problem of defining a surface except 
for a projective transformation by means of differential forms. The analytic 
basis for the study of any configuration by this method is a system of differ- 
ential forms, which define the configuration except for a projective trans- 
formation. The absolute calculus of Ricci may be employed in dealing with 
these forms. But when it is a question of finding the coordinates of a vari- 
able element of the configuration defined by the forms, the differential equa- 
tions that Wilczynski would use as fundamental appear. 

The reader who is familiar with the metric differential geometry of sur- 
faces in ordinary space will see that the two methods now under considera- 
tion could be used in developing this theory. Wilczynski would start with 
the differential equations of Gauss for the coordinates of a variable point on 
a surface and the direction cosines of the normal to the surface at the point. 
The condition of Gauss on the fundamental coefficients and the two equa- 
tions of Codazzi would enter the theory as conditions of complete integra- 
bility of the differential equations, and the first and second quadratic 
differential forms would appear later. Fubini would start, as Gauss did, 
with the quadratic differential forms as fundamental and would afterward 
obtain the differential equations, the conditions of Gauss and Codazzi, and 
the fundamental theorem on the determination of a surface when the six 
fundamental coefficients are given. 

Each method has its own advantages. Their starting points are different 
but the two theories soon become interlaced. Throughout most of this 
book a modified form of the American method will be used, the computa- 
tion of complete systems of invariants and co variants being omitted. The 
reader who is interested in this phase of the theory may consult the original 
memoirs cited hereinafter, and anyone interested in the Italian method 
may refer to Section 59 in Chapter VIII and to the treatise* by Fubini and 
Cech. 

The theory of non-developable surfaces in ordinary space is much simpli- 
fied by taking the asymptotic curves as parametric. Moreover, the funda- 
mental system of differential equations is susceptible of various canonical 
forms, each of which has its own advantages. The canonical form used by 
Wilczynski is analytically simple but lacks certain invariantive properties 
possessed by the canonical form of Fubini. The latter form will be used 
throughout this chapter unless the contrary is expressly stated, in spite of 
the fact that ruled surfaces are thereby excluded from consideration. The 

* Fubini and Cech, 1926. 1 and 1927. 1. 



66 PROJECTIVE DIFFERENTIAL GEOMETRY 

connection between Wilczynski's and Fubini's canonical forms is discussed 
further in Exercise 2 and in Section 60. 

The contents of the present chapter may be outlined as follows. After 
establishing an analytic basis for the theory of surfaces referred to their 
asymptotic curves in ordinary space, power series expansions are calculated 
for the local coordinates of a point on the surface, and some immediate 
geometrical applications thereof are made. Following the first two sections 
thus occupied come eight sections each of which is devoted to some phase 
of the general subject. Some of the many quadrics that have been associated 
with a point of a surface are discussed. Then the general theory of pairs of 
rectilinear congruences in a certain reciprocal relation with respect to a 
surface unifies the discussion of several particular covariant pairs of such 
congruences; this leads to the definition of a pair of flat pencils, called 
canonical pencils, covariantly associated with each point of a surface. Some 
important types of families of curves on a surface are considered under the 
headings of conjugate nets and of hyper geodesies. Cech's transformation, 
which comes next, is a point-plane correspondence associated with a point 
of a surface. Some more families of curves on a surface are considered under 
the general title of pangeodesics, and this chapter closes with a discussion 
of a certain envelope associated with a surface; this envelope is the locus 
of the vertices of a tetrahedron called the tetrahedron of Demoulin. 

16. The differential equations of a surface referred to its asymptotic net. 

The first problem considered in this section is to establish the defining system 
of differential equations of a surface referred to its asymptotic net. Then one 
of the integrability conditions of these equations enables us to reduce the 
equations to Fubini's canonical form by means of a transformation of the 
dependent variable. The integrability conditions for this form can be 
written very simply by means of certain useful formulas. This section 
closes with the determination of the residual group of transformations 
leaving Fubini's canonical form invariant, and the calculation of the effect 
of this group of transformations on the coefficients of the differential equa- 
tions. 

The defining system of differential equations that are fundamental in 
this chapter may be established in the following way. Let us consider an 
analytic non-developable proper surface S with the parametric vector 
equation x = x(u, t>) immersed in ordinary space $ 3 ; and let us suppose 
that the asymptotic net on S is parametric. Equations (II, 9) show that 
under this assumption we have 



16] SURFACES IN ORDINARY SPACE 67 

where L, Af, N are defined in equations (II, 10). Since the columns of a 
vanishing determinant are linearly dependent, it follows that the four co- 
ordinates xofa general point on the surface S are solutions of a system of equa- 
tions* of the form 



(1) {*" 

lav 



wherein the coefficients are scalar functions of u, v, and subscripts indicate 
partial differentiation. Moreover, the coordinates x are not solutions of any 
equation of the form 

(2) Ax uv +Bx u +Cx v +Dx = Q 

whose coefficients are scalar functions of u, v not all zero. We may remark 
that the asymptotic character of the parametric curves is rendered appar- 
ent by the form of equations (1) if the definition of an asymptotic curve is 
kept in mind. Moreover, singular points on the surface S at which an 
equation of the form (2) may happen to be conditionally satisfied will be 
avoided. 

The coefficients of system (1) are not arbitrary, but must satisfy certain 
partial differential equations, called integrability conditions, which we pro- 
ceed to discuss. The four third derivatives of x can be calculated from 
system (1), and can be expressed uniquely as linear combinations of x, 
x U j x v , x uv by differentiation and substitution. But of the five fourth deriva- 
tives of x there are three that can be calculated in two ways, namely, x uuuv , 
XUUM, Xuvw,' however, since the order of differentiation is immaterial, we 
must have 



The first and third of these equations are formal identities. But after some 
calculation the second equation reduces to an equation of the form (2), 
and then all four of its coefficients must be zero. Thus we obtain four 
integrability conditions on the coefficients of system (1). Of these conditions 
the only one that is needed immediately is the one that results from the 
vanishing of the coefficient of x uv . Neglecting for the moment the other 
three conditions and calculating only this one, we find it to be 



Wilczynski, 1907. 1, p. 244. 



68 PROJECTIVE DIFFERENTIAL GEOMETRY 

It follows from this condition that there exists a function 6 defined, except for 
an arbitrary additive constant, by the differential equations 

B u = o* ) 6 V = d . 

When the four integrability conditions are satisfied, it turns out that every 
derivative of x can be expressed uniquely as a linear combination of x, x u , 

Xy, Xuv 

It is clear that the surface S with which we started is not the only integral 
surface of system (1) ; in fact, any surface into which S can be projected is 
equally an integral surface of (1). Conversely, system (1) with the four 
integrability conditions satisfied defines a surface in ordinary space except for 
a projective transformation, since the theory of differential equations tells us 
that system (1) in the presence of the integrability conditions has four 
linearly independent solutions forming a fundamental set, and that the most 
general solution of (1) is a linear combination of these four solutions with 
constant coefficients. Such a system of partial differential equations is said 
to be completely integrabk. 

We shall next show how, by a suitable choice of proportionality factor 
for the homogeneous coordinates, to obtain Fubini's canonical form of 
system (1), and shall afterward write the integrability conditions explicitly 
therefor. The effect of the transformation 

(3) x = \(u, v)x (\ scalar) 

on system (1) is to produce another system of equations of the same form 
whose coefficients, indicated by dashes, are given by 



(4) 

7 = 7, = 0-2 log X, 

where is determined except for an arbitrary additive constant. The form 
of equations (1) shows that the u-curves (one family of asymptotics) on the 
surface S are straight lines if = 0, and the v-curves (the other family of 
asymptotics) are straight lines ify = Q. If we assume that the surface S is not 
ruled, so that 077*0, and if we choose X so that 

(5) 



16] 



SURFACES IN ORDINARY SPACE 



69 



then = log 187. Thus we obtain Fubini's canonical form of the differential 
equations of a non-ruled surface referred to its asymptotic curves, namely, 



/flx 
(o) 



I x vv = qx+yx u +6 v x v 



(6 = log 



We shall use this canonical system of equations as fundamental throughout 
this chapter unless otherwise indicated. The coordinates x, when multiplied 
by the factor X defined by equation (5), are called* Fubini's normal coordi- 
nates. 

The coefficients of system (6) satisfy three integrability conditions which 
we proceed to write. The formulas for the third derivatives of x expressed 
as linear combinations of x } x u , x v , x uv are found from system (6) to be 



(7) 

X vvv == v 

where TT, x are defined by the formulas 

(8) 

and <f, \f> by 

(9) 



+yx uv , 



The integrability conditions of equations (6) can now without difficulty be 
shown by the method explained above to be 



(10) 



If two functions Z, m are defined by placing 



Fubini, 1918. 1, p. 1033. 



70 PROJECTIVE DIFFERENTIAL GEOMETRY 

the integrability conditions (10) can be written in the form 



The most general transformation (3) which preserves Fubini's canonical 
form has X = const., as can be verified on inspection of equation (5). But any 
transformation of the independent variables that leaves the asymptotic net para- 
metric also leaves Fubini's canonical form invariant. In fact, aside from a 
mere interchange of the two asymptotic families, effected by a transforma- 
tion u = v,v = u y the most general transformation of parameters leaving the 
asymptotic net invariant is of the form u U(u), v = V(v). Let us consider 
the total transformation 

(11) x = cx , u=U(u) , v=V(v) (c = const.; cf/'FVO) . 

The effect of this transformation on the coefficients of equations (6) is found 
to be given by 

cp = p/U' 2 , 6 = 8V'/U' 2 , 

(12) \ . 

lg =$ 



where 6 is as usual determined except for an arbitrary additive constant. 
It is now easy to verify that 6 = log /fy and thus to show that Fubini's 
canonical form is preserved. Moreover, it could be shown that the trans- 
formation (II) is the most general transformation that leaves Fubini's canonical 
form invariant. 

17. The local coordinate system. Power series expansions. The study 
of the geometry of a surface in the neighborhood of one of its points P x is 
facilitated by the use of a local coordinate system intimately connected with 
the surface at this point. Power series expansions can be calculated for the 
homogeneous local coordinates of a point X near the point P x on the sur- 
face. Then one non-homogeneous local coordinate of the point X can be 
expanded as a power series in the other two coordinates. This series, ob- 
tained in the early part of this section, will be employed in the latter part 
for deriving some geometrical properties of the surface. In particular, the 
points called flecnode points will be defined, and their definition will be 
connected with the definition of flecnode points on ruled surfaces as em- 
ployed in Chapter II. 

Let us consider an integral surface S of equations (6). The points x, x u , 
x v , x uv are easily shown by simple calculations to be covariant under the 



1 17] 



SURFACES IN ORDINARY SPACE 



71 



transformation (11), and are ordinarily not coplanar. We shall use these 
points as the vertices of a local tetrahedron of reference at the point P x , with 
a unit point chosen so that any point whose coordinates in the original 
system are given by an expression of the form 



shall have local coordinates proportional to x\, . . . , #4. The effect of the 
transformation (11) on this local coordinate system is to change the unit 
point according to the formulas 

The unit point is therefore not ordinarily a covariant point. Two of the 
edges of the tetrahedron are the asymptotic tangents at the point P x . The 
complete geometric description of the tetrahedron will appear in Section 20. 




\ 



FIG. 9 



The coordinates of any point X near a point P x on a surface S as sketched 
in Figure 9 can be represented by Taylor's expansion as power series in the 
increments Au, At; corresponding to displacement on S from P x to the 
point X: 

It is possible by means of equations (6), (7) and the equations obtained 
therefrom by differentiation, together with the integrability conditions 



72 PROJECTIVE DIFFERENTIAL GEOMETRY 

(10), to express every derivative of x uniquely as a linear combination of 
x, x u , x v , x uv . Therefore X can be expressed uniquely in the form 



in which xi, . . . , 4 are power series in Aw, Ay which represent the local 
coordinates of the point X. Performing the calculations thus indicated 
leads to the following power series expansions of the local coordinates of a 
point near the point P x on the surface S referred to the tetrahedron x, x u , x v} x uv , 
with suitably chosen unit point: 



(15) 



The series (15) are fundamental in much of what follows. It is understood 
that the point X is sufficiently near to the point P x so that all the power 
series envisioned in this paragraph converge. 

Introducing non-homogeneous coordinates by the definitions 



(16) x 

we find at once, by use of (15), the following expansions: 



(17) 



z = 



From the expansions (17) it is possible to compute an expansion for z as a 
power series in x, y by setting z equal to a power series in x, y with unde- 
termined coefficients and then demanding that the expansions for x, y, z in 



17] SURFACES IN ORDINARY SPACE 73 

(17) shall satisfy this equation identically in Aw, At; as far as the terms of any 
desired order. Thus we find* to terms of the fourth order 



( . 
^ ' 



By way of application of this expansion let us study the intersection of 
the surface S and the tangent plane, z = 0, at a point P x of S. This intersec- 
tion is a plane curve C. Referred to the triangle x, x u , x v the equation of 
the curve C is the right member of equation (18) set equal to zero. Inspect- 
ing the terms of lowest degree therein we see that the tangent plane at an 
ordinary point of a surface intersects the surface in a curve with a node at 
the point, the nodal tangents being the asymptotic tangents of the surface at the 
point. This node is ordinarily a simple double point. To demonstrate the 
truth of the last statement let us expand y as a power series in x along 
the branch of the curve C which is tangent to the w-tangent, z y = 0. 
The result is 



Therefore this branch of the curve C ordinarily has only two consecutive 
points on the ^-tangent. A similar argument holds for the other branch of C. 

Let us drop for a moment the restriction 6 = log 7 which is characteristic 
of Fubini's canonical form of the fundamental differential equations. The 
necessary calculations would show that &<p is to be replaced by 2j30 u f$ u in 
the coefficient of x 4 in the expansion (18). When /3 is not identically zero, 
the conditional equation /3 = ordinarily defines a curve on the surface S. 
It is immediately seen that every point P x on this curve is ordinarily an 
inflexion point on that branch of the plane curve C (of intersection of S 
and the tangent plane at P x ) which is tangent to the u-tangent at P x . 
At such a point the node on the curve C is therefore & flecnode, i.e., both an 
inflexion and a node. A point P x on the surface S where the curve C possesses 
this singularity is called a flecnode point, or sometimes simply a flecnode. 
The curve = is called a flecnode curve, because every one of its points is a 
flecnode point. Similarly, with u and v interchanged, the curve y = is also 
a flecnode curve on the surface S. By equation (18) we now easily prove 
the following theorem, which the foregoing discussion suggests : 

An asymptotic tangent ordinarily intersects a surface S in three coincident 
points at its point of contact. But at a point of the flecnode curve & 0, the 
asymptotic u-tangent, z = y = Q, intersects S in four coincident points; simi- 
larly for the v-tangent and the curve 7 = 0. 

* Lane, 1927. 10. 



74 PROJECTIVE DIFFERENTIAL GEOMETRY 

In order to connect the definition of flecnode points formulated in the 
last paragraph with the definition of flecnode points stated in Section 13, 
let us consider the canonical form of equations (II, 20) for which p lk = Q 
(i, k 1, 2). It is easy now to verify that the x defined by equation (II, 14) 
satisfies a system of equations of the form (1) ; in fact this system is 

Xuu U j 

x vv = ( 

in which v is the t of Chapter II. Therefore the flecnode curves on a ruled 
surface, as defined in Section 13, are identical with the curve 7 = when is 
identically zero, in the notation of the present section. The tangent plane at a 
point of a flecnode curve on a ruled surface cuts the surface in the generator 
through the point and in a curve with an inflexion at the point. The entire 
intersection consequently has a node as well as an inflexion at the point, 
which is customarily and quite correctly called a flecnode. But since every 
point on a straight line is an inflexion point, a flecnode on a ruled surface 
is more precisely a biflecnode, that is, a node which is an inflexion point on 
both branches of the curve. 

Any plane, except the tangent plane, containing an ordinary asymptotic 
tangent at a point of a surface intersects the surface in a curve having an inflex- 
ion at the point. To prove this statement let us write the equation of a plane 
through the w-tangent, z = y 0, at a point P x of the surface S in the form 



The equation of the cone with its vertex at the point x uv and containing the 
curve C" of intersection of the surface 8 and this plane is found, by eliminat- 
ing z, to be 



Solving this equation for y as a power series in x, and then cutting this cone 
by the tangent plane, 2 = 0, we obtain the equations of a curve, 

2 = 0, 2/= 

This curve in the tangent plane has an inflexion at the point P x when is not 
zero, and consequently so does the curve C', as was to be proved. For this 
reason the asymptotic tangents of a surface are sometimes called inflexional 
tangents. 



18] SURFACES IN ORDINARY SPACE 75 

18. Quadrics of Darboux. There is a three-parameter family of quadric 
surfaces having contact of the second order at a point of an analytic sur- 
face, and among these quadrics there is a one-parameter family called the 
quadrics of Darboux which are distinguished from the others by a certain 
property which will be explained presently. Although common usage asso- 
ciates the name of Darboux with these quadrics, it seems that they were 
first considered by Hermite. Among the quadrics of Darboux there are 
some covariant quadrics of particular interest. Perhaps the most important 
of these are the quadric of Lie and the quadric of Wilczynski. Bompiani and 
Kloboucek have independently defined certain generalizations of the quad- 
ric of Lie which are called by Bompiani asymptotic osculating quadrics and 
which it seems appropriate to discuss in this section, although they are not 
themselves quadrics of Darboux. 

It is useful to state here a few pertinent facts from the general theory 
of the contact of an algebraic surface and an analytic surface. An algebraic 
surface A is said to have contact of order k with an analytic surface S, at an 
ordinary point P x on S, in case every curve on the surface S through the point 
P x has at P x exactly k-{-l consecutive points on the surface A. If then the 
series (15) are substituted in the local algebraic equation of the surface A, 
this equation is satisfied identically in Aw, &v as far as the terms of degree k. 
It is therefore (k+l)(k+2)/2 conditions for an algebraic surface to have con- 
tact of order k at a point of an analytic surface. If an algebraic surface A has 
contact of order k with an analytic surface S at a point P X1 then A inter- 
sects S in a curve* with a multiple point of order k+ 1 at P x , whose tangents 
lie in the directions obtained by setting equal to zero the terms of degree 
fc+1 that result when the series (15) are substituted in the equation of the 
surface A and the increments Aw, A# are replaced by the differentials du, dv. 

Since it is known that an algebraic surface of order h depends} on 
h (h 2 +6/iH-ll)/6 parameters, there is a limitation on the order of contact 
which it is possible for a given algebraic surface to have; this is expressed 
by the inequality (Jb+l)(A;+2)/2^(#+6A+ll)/6. For example, a plane 
is of order one and depends on three parameters, so that the tangent plane, 
4 = 0, at a point P x of a surface S is uniquely determined by the three condi- 
tions that it have contact of the first order with S at P x . 

A quadric surface, being of order two, depends on nine parameters. Since 
contact of the second order imposes six conditions, it follows that there are 
oo 3 quadrics having contact of the second order with a surface S at a point P x . 
The equation of a general one of these is obtained by writing the equation 

* Chasles, 1870. 1, p. 354. Letter from Moutard to Poncelet dated 1863. 
t Snyder and Sisam, 1914. 1, p. 206. 



76 PROJECTIVE DIFFERENTIAL GEOMETRY 

of the most general non-singular quadric and demanding that the series 
(15) satisfy this equation identically in AM, At; as far as the terms of the 
second degree. The result can be written in the form 

(19) x 2 X3+Xi(-Xi+k2X 2 +k3Xa+k4Xt) = Q , 

where the coefficients & 2 , kz,k^ are arbitrary, being constants as long as the 
point P x is fixed, and functions of u, v when P x varies over S. This quadric 
cuts the surface S in a curve with a triple point at P x , whose tangents are 
in the directions satisfying the equation 

(20) Pdu*+3k 2 du 2 dv+3k 3 dudv*+ ydv* = . 

There are certain curves called the curves of Darboux and others called 
the curves of Segre, which play a very significant role in the geometry of 
surfaces in ordinary space, and which will now be introduced. It is a simple 
matter to verify that if the binary cubic differential form that appears in 
equation (20) is a perfect cube of a linear form, then this form is one of the 
three linear factors of the cubic form (3du 3 +ydv 3 , and moreover & 2 , & 3 have 
one of the three pairs of values given by 

(21) /C 2 = e(^7) 1/3 , /c 3 = e 2 OV) 1/3 (* 3 =1) - 

A geometrical consequence is that if the three directions that satisfy equation 
(20) coincide at all, they coincide in one of the triple of directions satisfying the 
equation 

(22) 0dw 3 +7^ 3 = 0. 

These are by definition the directions* of Darboux, and the tangents in these 
directions are the tangents of Darboux, which were called by him tangentes 
d j osculation quadrique. The curves defined by the curvilinear differential 
equation (22) on the surface S are called the curves of Darboux, being en- 
veloped by the tangents of Darboux. A characteristic geometrical property 
of these curves may be stated as follows: 

The curves of Darboux on a surface S in ordinary space are the three one- 
parameter families of curves such that at each point P x of S the three tangents 
of these curves are in the directions in which may coincide the three triple-point 
tangents of the curve of intersection of S and a quadric having second-order 
contact with S at P x . 

* Darboux, 1880. 2, p. 356. 



18] SURFACES IN ORDINARY SPACE 77 

The directions* of Segre at a point P x of a surface S in ordinary space are 
defined to be the harmonic conjugates of the directions of Darboux with 
respect to the asymptotic directions. The tangents of Segre are by definition 
the tangents in the directions of Segre. The curves of Segre are defined to be 
the curves enveloped by the tangents of Segre, and their differential equa- 
tion is 

(23) pdu*-ydv* = Q . 

The quadrics of Darboux, which it is the principal purpose of the present 
section to study, can be introduced in the following way. Comparison of 
equations (22), (20) shows that any quadric (19) with ^2 = ^3 = and with k* 
arbitrary cuts the surface S in a curve whose triple-point tangents are the 
tangents of Darboux at the point P x . These oo l quadrics, represented by the 
equation 

(A; 4 arbitrary) , 



are called the quadrics of Darboux at the point P x of the surface S. A char- 
acteristic geometric property of these quadrics may be stated as follows: 

At a point P x of a surface S in ordinary space the quadrics of Darboux are 
the oo * quadrics each of which has second-order contact with S at P x and cuts S 
in a curve whose triple-point tangents are the tangents of Darboux of S at P x . 

Since contact of the third order imposes ten conditions, and since a 
quadric surface depends on nine parameters, it would seem possible to 
satisfy these conditions at particular points on a surface S. The method of 
counting constants suggests that it would be sufficient to impose one condi- 
tion on the parameters u, v; then there would be a curve of points at each 
of which a quadric having contact of the third order would exist. But this 
conclusion is not| correct. If the surface S is unspecialized, equation (20) 
shows that a point where there exists a quadric having contact of the third 
order must have curvilinear coordinates satisfying the two conditions 
= y = 0, and the quadric at such a point is subject only to the restrictions 
fc 2 = fc 3 = 0. Thus the following conclusions are justified: 

A point on an analytic surface S at which there exists a quadric surface 
having contact of the third order is an intersection of the two flecnode curves on 
the surface S. At every such intersection the oo 1 quadrics of Darboux have 
contact of the third order with the surface S. 

Several covariant quadrics of Darboux have been defined by various 
geometers. One of the most important of these quadrics is the quadric of 

* Segre, 1908. 1, p. 410. 
t Hermite, 1873. 1, p. 149. 



78 



PROJECTIVE DIFFERENTIAL GEOMETRY 



Lie, which is defined as follows. From one of the families of asymptotic curves 
on a surface S let us select a curve C, and on C a point P x . At P x and two neigh- 
boring points Piy PI on C let us construct the tangents of the asymptotic curves 

of the other family. These three asymptotic 
tangents determine a quadric surface. The 
limit of this quadric, as PI, P 2 independ- 
ently approach P x along the curve C, is* the 
quadric of Lie at the point P x of the sur- 
face jS. The equation of the quadric of 
Lie will show that it is a quadric of 
Darboux and that it remains the same 
if the roles of the two families of asymp- 
totic curves are interchanged in its defi- 
nition. The i6-tangent at a point P x is 
determined (see Fig. 10) by x, x u ; and the u-tangent at a point X, near P x 
and on the asymptotic v-curve through P X) is determined by X, X u repre- 
sented by the expansions 

X = x+x v Av+x vv &v 2 /2+ , 




p IG 



Any point Y on the latter tangent can be defined by a linear combination of 
the form 



(24) 



Y=hX+kX u 



(h, k scalars) . 



We find, on making use of equations (1), (7), that Y can be expressed in the 
form 



where the local coordinates xi, , z 4 of the point Y are given by 



(25) 



Demanding that the equation of a general quadric be satisfied by the series 
(25) for Xij . . . , x 4 identically in h, k, and identically in At; as far as the 



* Lie, 1922. 1, p. 718. Letter from Lie to Klein dated December 18, 1878. Wilczynski, 
1908. 2, p. 83. 



18] SURFACES IN ORDINARY SPACE 79 

terms of the second degree, we obtain the equation of the quadric of Lie at the 
point P x of the surface S, namely, 

(26) 2(0:20:3-0:10:4) - (/3y+<Uo: 2 4 = . 

That the quadric of Lie is covariant to the surface S, or is independent of 
the analytic representation used, is evident from its geometric definition. 
Moreover, this fact can be demonstrated analytically by showing by means 
of equations (12), (13) that equation (26) is absolutely invariant under the 
transformation (11). 

We shall next consider two quadrics associated with a point of a curve on 
a surface in ordinary space, which we shall denote by Q u and Q v respectively, 
and which we shall call, with Bompiani, asymptotic osculating quadrics. 
The asymptotic osculating quadric Q u at a point P x of a curve C on a surface S 
is* defined to be the limit of the quadric determined by three asymptotic u- 
tangents constructed at P x and at two neighboring points of C as these two points 
independently approach the point P x along the curve C. If C is the asymptotic 
p-curve through the point P XJ then the quadric Q u is the quadric of Lie, and 
in this sense Q u is a generalization of the quadric of Lie. The second asymp- 
totic osculating quadric Q v can be defined similarly. 

In order to find the equation of the quadric Q M , let us regard the curve C 
as imbedded in the one-parameter family of curves defined on the surface S 
by the equation 

(27) dv-\du = , 

where X is a function of u, v. The w-tangent at a point X near the point P x 
on the curve Cas determined by X 7 X u represented by the expansions 

X = 

X u = 

where x'=x tt +xx v . 

Any point Y on this tangent is defined, as before, by an equation of the 
form (24), but now we find for the local coordinates Xi, . . . , x* of the point 
Y the expansions 



11 Bompiani, 1926. 2, p. 263; Kloboueek, 1926. 11, p. 342. 



80 PROJECTIVE DIFFERENTIAL GEOMETRY 

Demanding that the equation of a general quadric be satisfied by the series 
(28) for xi, . . . , 4 identically in h, k, and identically in Aw as far as the 
terms of the second degree, we obtain the equation of the quadric Q u , 



(29){ 2X8( * 2X 



This generalization of the quadric of Lie is obviously not itself a quadric of 
Darboux, but it does have contact of the second order with the surface S 
at the point P x . 

The equation of the quadric Q v can be obtained in a way similar to the 
foregoing, or else can be written immediately by applying to equation (29) 
the substitution 

A* 2 ft p <p \ V \ 

(6()) (v 3 7 q t 1/X -X'/XV ' 

the result is 

f-X'-TX 3 



(31) { 



The osculating plane of the curve C defined by equation (27) at a point 
P x of a surface is determined by x, x', x", where 

(32) 



The local equation of this plane is found by a simple calculation to be 

(33) 2X(X*2-Z3) + (X'+0-0 tt X+<U 2 

hence its local coordinates are given by 

(34) ! = , 2 = 2X 2 , ? 8 = -2X , 



Another covariant quadric of Darboux is Wilczynski's canonical quadric, 
which was rediscovered by Bompiani in the following way. * If the curve C 
has an inflexion at the point P XJ i.e., has three consecutive points in its 
tangent line at P x , then its osculating plane (33) is indeterminate so that 
at the point P x we have 1 = 2 = 3 = 4 = 0, and therefore X = 0, X'= 0. 
Consequently the curve C is tangent to the asymptotic w-curve at P x , and 

* Bompiani, 1927. 11, p. 188. 



19] SURFACES IN ORDINARY SPACE 81 

the asymptotic osculating quadric Q v at the point P x of the curve C has the 
equation 

6 uv x\ = . 



This is* Wilczynski' s canonical quadric, and was defined by him in an en- 
tirely different way which we shall discuss in Chapter VIII, Section 60. 
Geometrically, the two asymptotic curves at a point play symmetric r61es 
in spite of the fact that the v-curve is not represented by equation (27). 
The following theorem summarizes these results: 

// a curve on a surface in ordinary space has an inflexion point, it is tangent 
to an asymptotic curve at the point, and the asymptotic osculating quadric de- 
termined by the asymptotic tangents of the other family is the canonical quadric 
of Wilczynski at the point. 

19. Reciprocal congruences. In 1916 G. M. Green called attention to a 
relation which may exist between two rectilinear congruences whose genera- 
tors are in correspondence with the points of a given surface. He called 




FIG. 11 

this relation the relation R. It can be defined as follows. Let us consider an 
unspecialized net N of curves on a surface S. Through a point P of the 
surface S there pass two curves Ci, C 2 of the.net N, with tangents ti, t% at the 
point P, as drawn in Figure 11. The tangents to the curves of the family of 
the net N that contains the curve Ci, constructed at the points of the 
curve 2, form a non-developable ruled surface JKi; similarly a ruled surface 

* Wilczynski, 1908. 2, p. 112. 



82 PROJECTIVE DIFFERENTIAL GEOMETRY 

Rz can be circumscribed about the surface S along the curve C\. Let us con- 
struct two planes TTI, ?r 2 through the point P, distinct from the tangent 
plane TT of S at P, and containing respectively the tangents ti, t 2 . These 
planes intersect in a line li which passes through the point P but does not lie 
in the tangent plane TT. The plane -K\ touches the ruled surface R\ at a point 
Pi y and similarly ?r 2 is tangent to R 2 at a point P 2 . The line 1 2 joining Pi, P 2 
lies in the tangent plane TT but does not pass through the point P. The two 
lines li, Z 2 are said to be in the relation R with respect to the net N. As the point 
P varies over the surface S, the lines li, h generate two congruences FI, F 2 
which are also said to be in the relation R with respect to the net N. 

When the net N is the asymptotic net on the surface S, the lines l\, l z are 
reciprocal polar lines with respect to the quadric of Lie at the point P of the 
surface S (see Ex. 34). This means that the polar planes of two and hence 
all points on the line l\ with respect to the quadric of Lie contain / 2 , and 
vice versa. Then the lines li, 1%, and likewise the congruences FI, F 2 , are 
simply said to be reciprocal with respect to the surface 8. The first problem 
is to determine the developables and focal surfaces of two reciprocal con- 
gruences, in order to prepare the way for the study of particular pairs of 
reciprocal congruences in the next section. 

Let us consider an integral surface S of equations (6), with the parametric 
vector equation x = x(u, v)\ a point P x on S; and any line l\ through P x but 
not in the tangent plane of S at P x . Such a line may be regarded* as de- 
termined by the point P x and the point P v defined by 

(35) y = ax u bx v +x uv , 

where the coefficients a, b are scalar functions of u, v. As u, v vary, the line l\ 
generates a congruence Fi. 

We shall immediately prove that the lines of the congruence FI can be 
assembled into a one-parameter family of developable surfaces in two ways 
(ordinarily determinate and distinct), so that there is one developable of each 
family through each line l\. We shall also show that the lines of the congruence 
FI are the common tangents of two surfaces (called focal surfaces of the con- 
gruence), each of the surfaces being the locus of one focus of the variable genera- 
tor li. These properties of the congruence FI are properties that are possessed 
by every rectilinear congruence, due account being taken of limiting 
cases. 

Let us consider a curve C, defined by the parametric equations u = u(t), 
v = v(t), through a point P x on a surface S, as shown in Figure 12. If the 

* Green, 1919. 1, p. 8G. 



19] 



SURFACES IN ORDINARY SPACE 



83 



ruled surface of the congruence FI that intersects the surface S in the curve 
C is a developable, and if the point P z defined by the formula 

(36) z = y+\x (X scalar) 

is the corresponding focal point of the line Zi, then h is tangent to the locus 




FIG. 12 



of the point P z as t varies, so that z' may be expressed as a linear combina- 
tion of x and y only. We find at once 



and after some calculation 



84 PROJECTIVE DIFFERENTIAL GEOMETRY 

where the functions A, B, F, G are defined by 



(37) { 



A = -a u - 

B= -b v -ab+pj+e uv , G = q-a v +ad v -a?+by . 



Setting equal to zero the coefficients of x u and x v in the expression thus ob- 
tained for z' as a linear combination of x, x u , x v , y, we have 



Eliminating X we obtain the differential equation of the curves in which the 
developabks of the congruence FI intersect the surface S, which will be called 
briefly the Ti-curves of the congruence: 

(39) (F-2ap+W)du*-(b v -a u )dudv--(G--2bv+>Y<f>)dv* = Q . 

The form of this equation shows that the Fi-curves ordinarily form a net, 
and that the lines of the congruence FI can then be assembled into oo 1 de- 
velopables in two distinct ways, so that there are two different developables 
of FI containing each line li. Moreover, eliminating the ratio dv/du from 
equations (38) we obtain 

(40) K+(A+B)\+AB-(F-2ap+M)(G - 267+7?) = . 

If Xi, X2 are the roots of this equation, the corresponding points zi, z% given 
by the formula (36) are the focal points of the lines h. The locus of each of 
the points 21, 22 is a focal surface of the congruence IV The line l\ is tangent 
to both of these surfaces, its points of contact being its two focal points. 
The reader may complete the discussion in the case in which the discrimi- 
nant of equation (39) vanishes, and the case in which this equation is in- 
determinate, showing that in the first case the congruence FI consists of one 
family of the asymptotic tangents of a surface, and that in the second case 
the congruence FI is a bundle of lines. 

We now start with a congruence Fi and arrive at a reciprocal congruence 
F 2 by a geometrical construction. Let us consider a line h determined by 
a point P x and the corresponding point P v . The polar plane of the point P x 
with respect to the quadric of Lie (26) is the tangent plane, x = ; the polar 
plane of the point P v has the equation 



19] SURFACES IN ORDINARY SPACE 85 

These two planes intersect in a line fa which has the equations 

(41) Xi 



and which crosses the asymptotic tangents at the points with local coordi- 
nates ( 5, 1, 0, 0), (a, 0, 1, 0). In the original coordinate system these 
are the points P p , P ff given by 

(42) p = x u bx , a x v ax . 

Clearly, the line fa lies in the tangent plane and does not pass through the point 
P x . As u, v vary, the line fa generates a congruence F 2 of lines some one of 
which lies in each tangent plane of the surface S but does not pass through 
the contact point. Conversely, the local equations of the polar planes of 
the points P p , P ff with respect to the quadric of Lie are respectively 

(43) Z8+&3 4 = 0, z 2 +az 4 = 0. 

These two planes intersect in the line l\ with which we started, so that l\, fa 
are reciprocal polar lines with respect to the quadric of Lie (or any quadric 
of Darboux), and FI, F 2 are in this sense reciprocal congruences. It is clear 
that if either of the congruences T\ and F 2 is given, the other is uniquely de- 
termined, so that it is possible to start with either congruence and from it 
arrive at the other by a geometrical construction. 

We shall now find the developables and focal surfaces of the congruence 
T 2 . If, as the pbint P x describes a curve C on the surface S, the line fa gener- 
ates a developable of the congruence F 2 , and if the point P f defined by the 
formula (see Fig. 12) 

(44) = p-fju<r (/z scalar) 



is the corresponding focal point of the line fa, then fa is tangent to the locus 
of the point P f , so that f ' may be expressed as a linear combination of p and 
a only. We find 



where 

p u = Fx (b 

<r v = Gx+yp-(a-0 v )<r , <r u = -(a u +ab)x-ap+x uv 



86 PROJECTIVE DIFFERENTIAL GEOMETRY 

Setting equal to zero the coefficients of x and x uv in the expression thus ob- 
tained for f ' as a linear combination of x, p, <r, x uv , we have 



(45) 



Elimination of /z gives the differential equation of the curves on the surface S 
corresponding to the developables of the congruence F 2 , which will be called the 
Tz-curves of the congruence: 

(46) Fdu 2 - (b v - a u )dudv - Gdv* = . 

Moreover, elimination of the ratio dv/du gives 
(47) 



If /xi, /x 2 , are the roots of this equation, the corresponding points ft, f 2 , given 
by the formula (44) are the focal points of the line Z 2 . Discussion of the 
special cases is left to the reader. 

We often find it convenient to use the locution a line li at a point of a 
surface to mean a line that passes through the point but does not lie in the 
tangent plane of the surface at the point. Similarly, a line lz at a point of a 
surface will mean a line that lies in the tangent plane at the point but does 
not itself pass through the point. The expressions a congruence Ti and a 
congruence T 2 are likewise used generically. The connotation of the sub- 
scripts 1 and 2 in this connection should be fixed in mind. 

Conjugate tangents and conjugate nets will now be defined. At a point 
of a surface in ordinary space two tangents are said to be conjugate, or to lie in 
conjugate directions, in case they separate the asymptotic tangents harmonically. 
A net of curves on such a surface is said to be a conjugate net in case the two 
tangents of the curves of the net at each point of the surface are conjugate tan- 
gents. It is not difficult to show that when the asymptotic net is parametric, 
conjugate directions dv/du have opposite signs. 

We conclude with a few remarks connecting the relation of conjugacy 
just now introduced with the theory of reciprocal congruences. The second 
of equations (45) shows that the conjugate of a tangent of a Tz-curve passes 
through the corresponding focal point of the line lz. Inspection of equations 
(39) and (46) makes it evident that, in the ordinary case, the Ti-curves and 
also the Tz-curves of two reciprocal congruences form conjugate nets if, and 
only if, b v a tt = 0. A congruence I\ whose developables intersect the funda- 
mental surface S in a conjugate net is said to be conjugate to S, and a con- 
gruence T 2 whose developables correspond to a conjugate net on the surface 



20] SURFACES IN ORDINARY SPACE 87 

S is said to be harmonic to S. If a congruence TI is conjugate to the surface 
S, the reciprocal congruence T 2 is harmonic to S, and conversely. 

20. The canonical pencils. Several covariant pairs of reciprocal congru- 
ences have been defined by various geometers in independent investigations 
of the geometry of surfaces. Among the covariant lines that have thus been 
associated with a point P of a surface S, perhaps the most interesting are the 
directrices of Wilczynski, the axes of Cech, the edges of Green, and the pro- 
jective normal of Green and Fubini. It is indeed remarkable that all of these 
lines just mentioned that pass through the point P lie in a flat pencil. This 
is called the first canonical pencil of the surface S at the point P. The recipro- 
cal lines also lie in a flat pencil called the second canonical pencil of the sur- 
face S corresponding to the point P. These pencils will be studied in this 
section. 

The history of the discovery of the projective normal is interesting. It is 
clear that the ordinary metric normal at a point of a surface is a line l\. 
The important role played by this normal in the metric differential geome- 
try of the surface suggested that it would be advantageous to have a pro- 
jectively defined substitute for this line. Moreover, it would be desirable 
to retain the property that the developables of the normal congruence inter- 
sect the surface in a conjugate net, and any other projective property that 
the metric normal might have. In 1916 Green announced* the discovery 
of the line now called the projective normal, which he called the pseudo- 
normal. His definition of this line is purely geometric, and his characteriza- 
tion is essentially that which will be explained later on in this section. In 
1918 Fubini published f his independent investigations on this subject. He 
arrived at the projective normal by a very elegant property which associates 
this line with the extremals of a projectively invariant integral, after the 
manner in which the metric normal is associated with the extremals of the 
arc-length integral, namely, the geodesies. This property will be explained 
in more detail in Section 22. Moreover, the property of the projective nor- 
mal that it is analytically the simplest covariant line li which generates a 
congruence TI that is conjugate to the surface S is also due to Fubini. 

The theory which we are now developing of the projective differential 
geometry of a surface in ordinary space may be observed to be self -dual, in 
the sense that it is invariant under the dualistic correspondence that con- 
verts each point of the surface into the tangent plane of the surface at the 
point (see Ex. 13). Reciprocal congruences correspond to each other in 
this duality. 

* Green, 1916. 4, p. 73; 1917. 2, p. 590; 1919. 1, p. 126. 
t Fubini, 1918. 1, p. 1038. 



88 PROJECTIVE DIFFERENTIAL GEOMETRY 

Comparison of equations (39) and (46) shows that, provided b v tt 5^0, the 
Ti-curves of a congruence coincide with the recurves of the reciprocal congru- 
ence if, and only if, 

(48) = ^/2, b = *>/2. 

The two reciprocal congruences thus defined, except in the case 6 V U = 0, 
are called the directrix congruences of Wilczynski. The lines li, Z 2 which 
generate them are denoted by d\, d 2 respectively, and are called the directrices 
of Wilczynski. The property used* by Wilczynski in defining these lines is 
that introduced in the next paragraph but one, and has the advantage of 
not requiring the restriction b v a u ?^0. 

We interpolate here a few remarks about line geometry preparatory to 
the demonstration to be given in the next paragraph. A linear congruence 
by definition consists of the lines that intersect two skew straight lines, 
which are called the directrices of the congruence. A special linear complex 
by definition consists of the lines that intersect one straight line, which is 
called the axis of the complex. It is left to the reader to demonstrate that 
the linear complex (I, 44) is special in case the coefficients a,-* in the equation of 
the complex satisfy the condition 

012034+013042-1- 01 4023 = , 

and that then the coefficients a lk are the coordinates of the axis of the complex. 
Moreover, two linear complexes intersect in a linear congruence. In fact, 

the two complexes determine a pencil of 
linear complexes all of which contain the 
congruence. The reader may show that 
in the pencil there are two special linear 
complexes whose axes are the directrices 
of the linear congruence. These ideas 
will be immediately developed in more 
FIG. 13 \ detail in a special case. A more extensive 

presentation of the elements of line 
geometry is found in Chapter VII of Wilczynski's book, f 

We shall now prove that the directrices di, d<t of Wilczynski at a point of a 
surface are the directrices of the linear congruence of intersection of the osculat- 
ing linear complexes of the two asymptotic curves at the point. Let us consider 
the asymptotic curve C through a point P x on a surface S, and let us con- 
sider also a point P v on C u near P x , as illustrated in Figure 13. The local 

* Wilczynski, 1908. 2, p. 114. f Wilczynski, 1906. 1. 




20] SURFACES IN ORDINARY SPACE 89 

coordinates yi, . . . , 3/4 of the point P v are found, by expanding x in powers 
of Aw and expressing x uu , . . . , x uuuu u as linear combinations of x, x u , x v , x uv , 
to be given by the power series 



(49) 



The local coordinates 21, . . . , 24 of a point P* on the tangent of the curve C u 
at the point P y are found similarly, by expanding x u in powers of AM, to be 
given by certain series which turn out to be precisely the derivatives of the 
series (49) with respect to Aw. Therefore the line coordinates o> t fc of the 
tangent of the curve C u at the point P v , as* these coordinates were defined 
in equation (I, 42), are given by the series 



(50) 



2= -0Aw 3 /3H 



Demanding that the equation (I, 44) of a linear complex be satisfied by the 
series (50) for *><* identically in Aw as far as the terms in Aw 4 , we obtain the 
equation of the osculating linear complex at the point P x of the asymptotic 
curve C u , namely, 

(51) W23 ^14 ^W34 = . 

Similarly, or by the substitution (30), the equation of the osculating linear 
complex of the asymptotic curve C v at P x is found to be 

(52) C023+W14 <pW42 = . 

All of the lines common to the two complexes (51), (52) are also in every 
complex of the pencil 

=0 , 



90 PROJECTIVE DIFFERENTIAL GEOMETRY 

where h, k are homogeneous parameters. In this pencil there are two special 
complexes; for them we find that /i = fc, so that their equations are re- 
spectively 

(53) 2cd23<pO)42 1/^34 = , 2&\\ <pC042 + ^W34 = . 



The two osculating complexes (51), (52) have in common all the lines of 
the linear congruence whose directrices are the axes of the special com- 
plexes (53). To obtain the equations in point coordinates of each of these 
directrices it is sufficient to write by means of the equations (I, 45) the con- 
ditions that must be satisfied by the coordinates of a point if the plane 
corresponding to it in the null system of each of the special complexes (53) is 
indeterminate. In this way, after a little calculation which will be omitted, 
we obtain again the directrices of Wilczynski, thus completing the proof. 
Let us pass to the definition of another pair of covariant reciprocal lines. 
It can be shown that at a point P x of a surface the osculating planes of the 
three curves of Segre intersect in a line h for which 

(54) a = */3, 6 = p/3. 

To make the demonstration one replaces X' by X M +X\z, in equation (33). 
One then replaces X by coX and again by w 2 X, where w is a complex cube root 
of unity. Thus one has three equations, in which the direction X of a curve 
of Segre has the value (ft/y) 1 / 3 , as equation (23) shows. Finally one takes 
suitable linear combinations of the three equations and reduces the result 
by means of the characteristic condition = log fiy and the definitions of <p, 
\l/ given in equations (9). These calculations are straightforward and ele- 
mentary, and the details will be omitted. This line* of intersection is now 
commonly called the first axis of Cech, although he himself called it the line 
of Segre. It and its reciprocal, the second axis, will be denoted by ai, a% re- 
spectively. 

The directrix d\ and the axis ai at a point P x on a surface S determine a 
plane through P x called f the canonical plane of the surface S at the point P x . 
The equation of this plane, shown in Figure 14, is 

(55) <px% ^z 3 = . 

The pencil of lines lying in this plane and having its center at P x is called 
the first canonical pencil of the surface S at the point P x , and any line of this 
pencil, joining P x (l, 0, 0, 0) to a point 

(0, fc^, k<p, 1) (k = const.) , 

* Cech, 1922. 2, p. 199. t Fubini and Cech, 1926. 1, p. 155. 



20] 



SURFACES IN ORDINARY SPACE 



91 



is spoken of as a canonical line of the first kind. The canonical plane (55) 
intersects the tangent plane, z 4 = 0, in a line called the first canonical tangent 
h of S at P x . 

The directrix d 2 and the axis a 2 which lie in the tangent plane at the point 
P x of the surface S intersect in a point called the canonical point of the sur- 
face S corresponding to the point P X) or briefly at P x . The local coordinates 
of this point are 0, \fr, <p, 0. The pencil of lines lying in the tangent plane and 




FIG. 14 



having its center at the canonical point is called the second canonical pencil 
of the surface S corresponding to the point P x , and any line of this pencil, 
crossing the asymptotic tangents at the points (kp, 1, 0, 0) and (k\[/, 0, 1,0) 
is spoken of as a canonical line of the second kind. The second canonical 
tangent fa by definition joins the point P x to the canonical point. It should be 
observed that the two canonical tangents are conjugate tangents. It should be 
further observed that a canonical line of the first kind and the canonical line 
of the second kind with the same value of k are a pair of reciprocal lines for which 

(56) a=k\fr, b=k<p (k = const. ). 

For the directrices of Wilczynski we have seen that fc= 1/2, and for the 
axes of Cech, k= 1/3. We shall now consider* the edges ei, e% of Green, 
* Green, 1916. 4, p. 73; 1917. 2, p. 591; 1919. 1, p. 114. 



92 PROJECTIVE DIFFERENTIAL GEOMETRY 

which may be defined analytically as those canonical lines for which 
k 1/4; and the projective normal n\ of Green and Fubini, which may be 
defined analytically as the canonical line Zi for which k 0. Finally, we shall 
show how any canonical line can be characterized by a certain cross ratio. 
It will first be shown, after the manner of Green's investigations, that the 
edges have the following characteristic geometric property. At each point 
P x of a surface S the edge e% crosses the tangent of each asymptotic curve in the 
pole of the other asymptotic tangent with respect to the four-point conies of the 
projection of the first asymptotic curve from any point on the edge e\ onto the 
tangent plane, and that this geometric relation holds for no other pair of recipro- 
cal lines. For the purpose of the proof let us make the following observa- 
tions. If the coordinates of a point referred to the tetrahedron z, z u , x v , x uv 
are 2/1, . . . , f/ 4 , and if the coordinates of the same point are Xi t . . . , z 4 when 
referred to the tetrahedron x, p, a, 2/+Xz, where p, a are defined by equa- 
tions (42) and y by (35), and X is an arbitrary scalar function of u, v, then 
the identity 



yields the equations for the transformation of coordinates between the two 
tetrahedrons; after solution for x\, . . . , x* these equations can be written 
in the form 






The parametric equations of the projection C' u of the asymptotic curve C u 
from the new vertex (0, 0, 0, 1) onto the tangent plane, z 4 = 0, are found by 
substituting the series (49) for y i} . . . , t/ 4 into equations (57) and taking 
such a linear combination of the resulting coordinates xi, . . . , # 4 and of 
0, 0, 0, 1 as will make the fourth coordinate vanish. These parametric equa- 
tions, to terms of as high degree as will be needed, are 






the fourth coordinate being zero in the remainder of this paragraph. It will 
be observed that equations (58) are the result of substituting the series (49) 
for 2/1, . . . , t/ 4 into the first three of equations (57) and neglecting the fourth 
equation. It will be observed further that, as far as written, these equations 



20] 



SURFACES IN ORDINARY SPACE 



93 



are independent of X and hence of the position of the center of projection on 
the line li that is now being used as the edge Xz = 3 = of the tetrahedron of 
reference. Imposing on the general equation of a conic the conditions that it 
be satisfied by the series (58) for x\, x 2) x s identically in Aw as far as the 
terms in Aw 3 , we obtain the equation of the four-point conies at the point P x of 
the projection Cu of the asymptotic curve C u , namely, 

where h is a parameter. The polar line of the point P p (0, 1, 0) with respect 
to any one of these conies (see Fig. 15) has the equation 




FIG. 15 

and this line coincides with the asymptotic v-tangent, 2 = 0, in case b = p/4. 
Similarly, with the roles of the asymptotics C u and C v interchanged, we get 
a = \l//4. Thus we obtain the edges of Green, as was to be proved. 

We next establish the following theorem, which states a characteristic 
geometric property of the projective normal defined analytically as the 
canonical line for which k = 0. The projective normal n\ is the harmonic con- 
jugate of the directrix di with respect to the edge e\ and the canonical tangent ti. 
This statement is immediately verified by observing that the cross ratio of 
four canonical lines is the cross ratio of the corresponding values of the con- 
stant fc, and that 

( - 1/4, . , - 1/2, 0) = - 1 . 



Green used essentially this property to define the projective normal. 



94 PROJECTIVE DIFFERENTIAL GEOMETRY 

We are now prepared to complete the geometric description of the local 
tetrahedron introduced in Section 17 at a point P x of a surface. We observe 
that the line xx uv is the projective normal, and that the line x u x v is its reciprocal 
n 2 . Thus we have the geometric definition of two more lines, besides the 
asymptotic tangents, of the tetrahedron, which was already known to be 
covariant. The vertices x u , x v are the intersection points of the asymptotic 
tangents and the line n 2 . The lines x u x uv , x v x uv are respectively the v-tan- 
gent and w-tangent of the surfaces generated by the points x u , x v . 

Setting a = 6 = in Section 19 we find that the Fi-curves of the projective 
normal congruence, which are called the projective lines of curvature, have 
the equation 



(60) 

while the corresponding IVcurves have the equation 

(61) 



Both of these nets of curves are conjugate nets. 

The cross ratio of the canonical tangent h, the projective normal n\, the 
directrix d\, and any canonical line Ci, which corresponds to a general con- 
stant k, has the value 2k. Thus the general canonical line c\ can be char- 
acterized by its cross ratio with three other lines each of which has already 
been characterized geometrically. 

21. Conjugate nets. The historical reason why conjugate tangents at a 
point of a surface are so named is that they can be defined as conjugate 
diameters of the Dupin indicatrix* of the surface at the point. Instead of 
this metric definition a projective definition of conjugate tangents was 
stated in the latter part of Section 19. 

It is known that there are infinitely many conjugate nets, as defined in 
Section 19, on a non-developable surface in ordinary space. In fact, when 
any one-parameter family of non-asymptotic curves is given on such a 
surface, there exists another one-parameter family of curves on the surface 
such that the two families form a conjugate net. When it is a matter of 
studying only a single conjugate net, it may be advantageous to choose 
this net for the parametric net, as will be done in the next chapter. But in 
solving problems concerning several, or infinitely many, conjugate nets on a 
surface, it is convenient to choose the asymptotic net on the surface as the 
parametric net. 

* Dupin, 1813. 1, p. 41. 



21] SURFACES IN ORDINARY SPACE 95 

When a conjugate net is given on a surface in ordinary space, there are 
two congruences of the types TI and F 2 determined by the net in a way that 
will be described presently and called respectively the axis congruence and the 
ray congruence of the net. These are of fundamental importance for the 
geometry of the net. Moreover, there is a one-parameter family of conjugate 
nets determined in a certain way by the given net on the surface and called 
a pencil of conjugate nets. At each point of the surface there are several loci 
associated with such a pencil. Among these, two of the most significant are 
a certain cubic curve in the tangent plane, called the ray-point cubic, and 
its dual called the axis-plane cone. These configurations will be studied in 
this section. 

The curvilinear differential equation of a conjugate net N\ on an integral 
surface S of system (1) can be written in the form 

(62) cfo 2 -XW = 0, 

where X is a function of u, v. There exists on the surface S a conjugate net 
uniquely determined by the net N\, with the property that at each point 
of S its tangents separate the tangents of the net N* harmonically. This net 
is called the associate conjugate net of the net N\ t and its differential equation 
is easily shown to be 

(63) dv 2 +\ 2 du 2 = Q . 

In fact, direct calculation verifies that the two tangents of the net (62) and 
the two tangents of the net (63) form a harmonic group, so that the relation 
between the two nets is entirely reciprocal. 

The axis congruence of a conjugate net is determined by the net in the 
following way. The axis* of a point P x on a surface S with respect to a con- 
jugate net N\ on S, is defined to be the line of intersection of the two osculating 
planes at P x of the two curves of the net N\ that pass through the point P x . This 
line is also called the axis of the net N\ at the point P x . Referring to Figure 16, 
let us denote the two curves of the net N\ that pass through the point P x by 
Cx and C- x according as the direction dv/du has the value X or X. At the 
point P x the osculating plane of the curve C x has the equation (33), while 
the osculating plane of the curve C_ x has the equation obtained therefrom 
by first replacing X' by X M + XX V and then changing the sign of X. Taking 
suitable linear combinations of these two equations, we obtain two equa- 
tions of the form (43) with a, 6 given by 

(64) a = [0 v +(log X) v +/3/X 2 ]/2 , 6 = [ft, -(log A) tt + 7 X 2 ]/2 . 
* Wilczynski, 1915. 1, p. 312. 



96 PROJECTIVE DIFFERENTIAL GEOMETRY 

Therefore the axes of all the points of a surface S with respect to a conju- 
gate net N\ on Sform a congruence Fi; this is called the axis congruence of the 
net JVx. 

The associate axis congruence of a conjugate net is by definition the axis 
congruence of the associate conjugate net. The associate axis congruence of 
the net N\ is also a congruence Fi, and for it the values of a, 6 are obtained 
from equations (64) by changing the sign of X 2 . 




FIG. 16 



It will now be shown that the tangents of the curves of one family of a con- 
jugate net constructed at the points of each fixed curve of the other family form a 
developable surface, and its edge of regression will be determined. The point 
P y defined by placing 

y = kx+Xu ^x v (k scalar) 

is on the tangent of the curve C_ x at the point P x . As the point P x varies 
along the curve C\, the point P v generates a curve whose tangent at P v is 
determined by P v and the point y' given by 

t/' = yu+X^ = (p+fc+fc,X-?X 2 )z+^^ 

This point is on the tangent of the curve C-\ in case k satisfies the equation 
(65) 2Xfc- 



21] SURFACES IN ORDINARY SPACE 97 

Therefore, as the point P x varies along the curve C x , the tangent of the 
curve C_x generates a developable whose edge of regression is generated by 
the focal point of the tangent of the curve C_ x , namely, the point 

(66) ( X '-0-0 u x+<U 2 +7X 3 )*+2X(z tt ->og . 

A similar argument can be made with the curves C x and C_ x interchanged. 
The ray congruence of a conjugate net is determined by the net in the fol- 
lowing way. The point (66) is called the ray-point of the curve C x , corre- 
sponding to the point P x , or sometimes simply the ray-point of C\ at P x . 
Its local coordinates are given by 



z 3 =-2X 

The ray-point of the curve C_\, corresponding to the same point P x , is ob- 
tained by changing the sign of X in equations (67). The line joining these 
two points is called the ray* of the point P x with respect to the net AT X , or some- 
times also the ray of the net N\ corresponding to the point P x , or simply the 
ray of N\ at P x . Taking linear combinations so as to eliminate first x v and 
then x u from the expression (66) and the expression obtained therefrom by 
changing the sign of X, we find that the ray crosses the asymptotic tangents 
at the points (42) with a, b given by 

(68) a = [0,+(log X) v -/3/X 2 ]/2 , b = [0 u - (log X) u + T X 2 ]/2 . 

Therefore the rays of all the points of a surface S with respect to a conjugate 
net N\ on Sform a congruence T 2 ; this is called the ray congruence of the net 
N\. Comparison of equations (64) and (68) shows that the ray congruence of 
a conjugate net is the reciprocal of the associate axis congruence of the net. 
Similarly, defining the associate ray congruence of a conjugate net to be the 
ray congruence of the associate conjugate net, we see that the axis congru- 
ence of a conjugate net is the reciprocal of the associate ray congruence of the net. 
We now define a pencil of conjugate nets. The class of 1 conjugate nets 
on a surface every one of which has the property that at every point of the surface 
its two tangents form with the tangents of a fundamental conjugate net the same 
cross ratio is called a pencil^ of conjugate nets. The differential equation of a 
general net N\h of the pencil p\ of conjugate nets determined by the net 
7V X whose equation is (62) can be written in the form 

(69) dv* - XWdu 2 =0 (h = const.) , 
* Ibid., p. 317. t Wilczynski, 1920. 1, p. 216. 



98 PROJECTIVE DIFFERENTIAL GEOMETRY 

since the cross ratio of the four values of dv/du calculated from equations 
(62) and (69) is constant. 

The curve called the ray-point cubic next claims our attention. In order 
to define this curve let us consider a point P x on a surface S, a pencil p x of 
conjugate nets on S, a net N\h of this pencil, and the curve C\h of this net 
that passes through the point P x . The local coordinates of the ray-point, 
corresponding to the point P x , of the curve C\h are found from equations 
(67), on replacing therein X by \h, to be given by 

- ft - 6 u \h + e v \ 2 h 2 



Homogeneous elimination of h from these equations gives the equations of 
the locus of this ray-point as the net N\h varies over the pencil p\, namely, 

(70) x, = Ix 2 x 3 - 0x1 - yxl = , 
where I is defined by 

(71) l = 2x!+[6 v - (log X)]o;2+[0,+(log X) Ja* . 

The locus of the ray-point, corresponding to the point P x , of the curve C-\h 
of the net N\h that passes through P x is found, by the foregoing argument 
with the sign of h changed, to be the same curve (70). This curve of the 
third order in the tangent plane of the surface S at the point P x is * called 
the ray-point cubic of the pencil p x , corresponding to the point P x . It has a 
double point at P x with the asymptotic tangents at P x for double point 
tangents (see Fig. 17). It has three points of inflexion which lie on the line 
x 4 = Z = 0, called the flex-ray of the pencil p x corresponding to the point P x , 
some one of the inflexions lying on each of the lines 

(72) x 4 = jtoS+73i = 0. 

These three lines are the tangents of Darboux } as can be seen by referring to 
equation (22) and by observing that the local equations of the tangent to a 
curve dv\du = at P x are 4 = 23 Xz 2 = 0. Comparison of the equations 
x 4 = Z = and (41) shows that the flex-rays of a pencil p x , corresponding to all 
the points of a surface, form a congruence T^for which 

(73) a = [,+ (log X) J/2 , b = [ft, - (log X) J/2 . 
* Lane, 1922. 3, p. 289. 



21] 



SURFACES IN ORDINARY SPACE 



99 



The reader may have observed that when the asymptotics are real there 
is only one real tangent of Darboux at each point P x , and likewise only one 
real tangent of Segre, and only one real inflexion of the ray-point cubic (70). 
We may remark further in this connection that the envelope of the rays 




. real 
Seyre tangent 



FIG. 17 

of all the nets of a pencil p x , corresponding to a point P x , is a conic called 
the ray conic (see Exs. 11, 25) which has interesting relations to the ray- 
point cubic. 

The dual of the ray-point cubic of a pencil of conjugate nets, correspond- 
ing to a point of a surface, is called the axis-plane cone of the pencil at the 
point. The axis-plane cone of the pencil p x at the point P x of the surface S is, 
therefore, the envelope of the osculating planes at the point P x of all the 



100 PROJECTIVE DIFFERENTIAL GEOMETRY 

curves of the pencil p x that pass through P x . The local coordinates of the 
osculating plane at the point P x of the curve CXA can be obtained from equa- 
tions (34) by replacing therein X by \h. Then homogeneous elimination of h 
gives the equations of the envelope of this osculating plane as the net N\h 
varies over the pencil p x , namely, 



(74) & 
where m is defined by 

(75) m = 2 4 -[0 M -(log X)J{.-[0.+(log 



The same equation (74) would have been obtained if the curve C_\A had 
been used instead of the curve C\A, and this is the equation of the axis-plane 
cone of the pencil p x at the point P x . This cone is of the third class and has 
its vertex at the point P x . It has the tangent plane of S at P x for bitangent 
plane; in fact, it touches this plane along the asymptotic tangents through 
P x . It has three cusp-planes which pass through the line i = ra = 0, called 
the cusp-axis of the pencil p x at the point P x , some one of the planes inter- 
secting the tangent plane in each of the lines 

(76) & = #8+7HO. 

These three lines are the tangents of Segre, as can be seen by observing that in 
point coordinates the equations of the tangents of Segre are z 4 =/3o| 70;! = 
0, and that for a line through P x we have & = 0, z 2 /f 3 = x s / ( 2 ) . The cusp- 
axes of a pencil p x , at all the points of a surface, form a congruence Ti for which 
a, b are given by equations (73), as the equations {i = m= ! =0 show. Hence we 
deduce the theorem: 

The flex-ray and cusp-axis congruences of a pencil of conjugate nets are 
reciprocal congruences. 

The curves (22) of Darboux and the curves (23) of Segre belong to a 
pencil of conjugate nets called* the Segre-Darboux pencil, for which X = 
(/3/7) 1/3 . The cusp-axis of the Segre-Darboux pencil is the axis a\ of Cech, 
since for this pencil equations (73) reduce to (54). Moreover, the flex-ray 
of the Segre-Darboux pencil is the second axis a 2 of Cech. 

22. Hypergeodesics. Union curves and planar systems. The curvesf 
defined on a surface S by a differential equation of the form 

(77) v"=A+Bv'+Cv'*+Dv'* , 

* Ibid., p. 293. 

f Fubini, 1918. 1, p. 1034; Wilczynski, 1922. 4. 



22] SURFACES IN ORDINARY SPACE 101 

in which the coefficients are functions of u, v and accents indicate total 
differentiation with respect to u, are called hyper geodesies, because the differ- 
ential equation of the geodesies in the metric theory of surfaces can be writ- 
ten in this form (see 48, Chap. VI). Since (77) is a differential equation 
of the second order, it follows that this equation defines a two-parameter 
family of hypergeodesics on the surface S. Within a suitably restricted 
region of S there is just one of these hypergeodesics through any two dis- 
tinct points. Moreover, a hypergeodesic is uniquely determined by a 
point on S and a direction at the point. Other properties of hypergeodesics 
will be discovered presently; and some special kinds of hypergeodesics will 
be considered, among which are the curves of a pencil of conjugate nets, the 
so-called projective geodesies connected with Fubini's definition of the pro- 
jective normal, the union curves of a congruence Ti, and planar systems of 
curves. 

The equations of the envelope of the osculating planes at a point P x of all 
the hypergeodesics (77) that pass through P* are found by replacing v' by X 
in equation (77), substituting the resulting expression for X' in equations 
(34), and eliminating X homogeneously. The result is 

(78) ? 1 = ^ 2 [2{4-(^-B){ 3 -(^+C)y-(/3+A)?I-( 7 -JD)J 2 3 = 0. 

This envelope is therefore a cone, which has properties similar to those of 
the cone (74). In particular it has a cusp-axis, which is called the cusp-axis 
of the hypergeodesics at the point P x , and which passes through the point 
(0, a, 6, 1) for which 

(79) . a=(e 9 +C)/2, b = (e u -B)/2. 

Dually, the locus of the ray-points, corresponding to the point P x , of all the 
hypergeodesics (77) that pass through P x is the curve 

(80), x,==x 2 x z [2x 1 -i-(e u -B)x 2 +(d v -{-C)x 3 l-(l3-A)xi-(y+D)xl = () . 

This curve is to be compared with the curve (70) ; it has a line of inflexions 
which is the reciprocal of the cusp-axis of the cone (78) and is called the 
flex-ray of the hypergeodesics at the point P x . 

Our first projective example of a family of hypergeodesics will now be 
adduced. If equation (69) is solved for h, and if h is then eliminated by 
total differentiation with respect to u, it becomes apparent that the curves 
of a pencil p x of conjugate nets constitute a family of hypergeodesics for which 

(81) 4=Z) = 0, J5=(logX), C=(logX) v . 



102 PROJECTIVE DIFFERENTIAL GEOMETRY 

The vanishing of the coefficients A, D signifies that the asymptotic curves 
are included in the family of hypergeodesics. Moreover in this example we 
obviously have B V = C U . 

In order to obtain another example of hypergeodesics, let us observe 
that the differential form (Bydudv is easily shown to be absolutely invariant 
under the transformation (11). Therefore the integral 



calculated along a curve is invariant. This integral is called the projective 
arc-length of the curve along which it is calculated. Making use of Euler's 
equation for the extremals of an integral of the form f<p(u, v, v')du, namely, 

(82) 0>,v V" = <p v ~ <f> UV > ~ Vvv'V' , 

we find that the differential equation of the extremals of the projective 
arc-length is of the form (77) with 

(83) A = Z) = 0, B = 6 U , C=-6 V . 

These curves are called the projective geodesies on the surface S. Reference 
to equations (79) makes it evident that the cusp-axis of the projective geo- 
desies is the projective normal. This is Fubini's definition* of the projective 
normal. 

Union curves^ are a third example of hypergeodesics. A curve on a surface 
is called a union curve of a congruence FI in case the curve is such that its 
osculating plane at each of its points P x contains the line l\ of FI through P x . 
The differential equation of the union curves of a general congruence FI is 
obtained by replacing X by v f in equation (33) and substituting for x\, . . . , x\ 
the coordinates 0, a, 6,1 respectively. The result is an equation of the 
form (77) with 

(84) A = -0, B = u -2b, C=-(0 v -2a), D = y . 

The envelope of the osculating planes at the point P x of all these curves 
that pass through P x is, of course, the line l\ of the congruence FI through P x , 
so that these planes form a pencil with Zi as axis. For this reason the union 
curves of a congruence are sometimes called! an axial system of curves. 

A curve on a surface S is called a dual union curve of a congruence F 2 in 
case the curve is such that its ray-point corresponding to each of its points P x 

* Fubini, 1918. 1, p. 1038. 

t Sperry, 1918. 2, p. 214. t Bompiani, 1923. 1, p. 268. 



22] SURFACES IN ORDINARY SPACE 103 

lies on the line Z 2 of F 2 in the tangent plane of S at the point P x . Using equa- 
tions (67) with X = */, and the equations (41) of a line 1 2 , we find that the 
differential equation of the dual union curves of a congruence F 2 is of the 
form (77) with 

(85) A=/3, B = 6 u -2b, C= -(6 v -2a) , =-7. 

A planar system of curves* on a surface is defined as follows. Let us con- 
sider a congruence F of non-singular quadric surfaces one of which is 
associated with each point P x of a surface S in such a way that it contains 
the asymptotic tangents through P z , but does not have contact of order as 
high as the second with S at P x . The equation of such a non-singular quad- 
ric can be written in the form 

(86) x<tX3-\-Xi(kiXi+kiXi+k z X3+kiXi)=Q , /Ci(l-f/Ci)^0 , 

in which fci, . . . , k* are constants as long as P x is fixed, and functions of 
w, v when P x varies. Then a planar curve of the congruence F is defined to be 
a curve on the surface S such that at each of its points P x its asymptotic osculat- 
ing quadric of one family (which is known to contain the asymptotic tan- 
gents through the point P x ) intersects the associated quadric of T in a residual 
pair of straight lines instead of in a residual non-singular conic as would 
ordinarily be the case. Using equation (86) and the asymptotic osculating 
quadric Q u with equation (29) we find, by eliminating Xi and setting the 
discriminant of the result equal to zero, that the differential equation of 
the corresponding family of planar curves of the congruence F is of the 
form (77) with' 

rjS(l-fci), = 2r/c 2 -(log T )u, C= - 



The reason why such a family of curves is called a planar system will be ex- 
plained in Exercise 17 of Chapter V. 

Using an equation of the form (88) but with coefficients I instead of k, and 
using the asymptotic osculating quadric Q v with equation (31), we obtain 
a second planar system with an equation of the form (77), for which 

fyA = -2 
I C=-2 

* Bompiani, 1926. 2, p. 263. 



104 PROJECTIVE DIFFERENTIAL GEOMETRY 

When the expressions for corresponding coefficients A, B, C, D in equations 
(87) and (88) are equated, the resulting equations can be solved uniquely 
for the coefficients I in terms of the coefficients k, and vice versa. Thus the 
following theorem of Bompiani is established : 

// a planar system is defined by one family of asymptotic osculating quadrics 
and a congruence of quadrics, then a second congruence of quadrics can be 
determined so that it and the other family of asymptotic osculating quadrics 
define the same planar system. 

23. The transformation of Cech. We begin with a definition. The trans- 
formation* represented analytically by the equations 

f i = , <r 2 = xzxl , (7{ 3 = 4*3 , 

I o- 4 = xix&3+k(flxl+vxl) (A; = const.) , 



where v is a proportionality factor, between planes with local coordinates 
through a point P x of a surface S and points with local coordinates x in the 
tangent plane of S at P x , is the transformation of Cech. Observing that x^ = 0, 
and solving equations (89) for the ratios of xi, #2, #3, we have the inverse 
transformation, 



/ pXl = - 

i 

I p^2 = Q& , px$ = 3$ i z 4 = , 

where p is a proportionality factor. It will be observed that the transforma- 
tion of Cech is not only a cubic birational transformation but is actually a 
Cremona transformation. It will be observed, further, that the transforma- 
tion involves an arbitrary constant k, which is independent of u, v. By giv- 
ing particular numerical values to k we see that at each point of a surface 
there are < 1 particular transformations of Cech. 

It is of interest to characterize geometrically the general transformation 
(89) when k is an arbitrary constant, and then to distinguish geometrically 
any particular one of these transformations from all the rest. This latter 
distinction is made by first defining geometrically two particular trans- 
formations (89), namely, the polarity of Lie and the correspondence of Segre, 
and then using a certain cross ratio. 

For the purpose of characterizing geometrically the general transforma- 
tion of Cech, let us consider the pencil of cubic curves 

(91) xt 

* Cech, 1922. 2, p. 192. 



23] SURFACES IN ORDINARY SPACE 105 

with the same constant k that appears in equations (89), and with arbitrary 
functions a, b of u, v. This pencil lies in the tangent plane at the point P x 
of the surface S. Every cubic of this pencil has a double point at P x , with 
the asymptotic tangents through P x for double-point tangents; and has 
three inflexions which lie on the line Z 2 of equations (41), some one of the in- 
flexions lying on each of the tangents of Darboux (72). These properties 
are characteristic of the pencil of curves. Now the general transformation 
of Cech has the following properties. It is a one-to-one correspondence be- 
tween points in the tangent plane, #4 = 0, and planes through the contact 
point P x (l, 0, 0, 0) such that to every plane (0, 2 , 3, 4) through P x dis- 
tinct from the tangent plane corresponds a point (xi, x 2 , x 3 , 0) on the line 
x* = 2X2 323 = conjugate to the line of intersection, #4 = 2^2 +323 = 0, of 
the given plane and the tangent plane. Moreover, to the planes of a pencil 
with any line l iy joining the point P x (l, 0, 0, 0,) to a point (0, a, 6, 1), 
as axis correspond the points on a cubic of the pencil (91), the line of in- 
flexions of the cubic being the line Z 2 reciprocal to li. Conversely,* every 
transformation having these properties can be shown to be a transforma- 
tion of Cech, by a simple argument which will be omitted. Thus the 
general transformation of Cech is characterized. 

Of the several particular transformations of Cech which have received 
attention at various times in the past, we shall mention here only two. 
First, the transformation for which fc = is without difficulty recognized to 
be the correspondence between points in the tangent plane, x^ =0, and their 
polar planes with respect to the quadric of Lie, represented by equation (26) . 
This transformation between the points and planes of space is sometimes 
called the polarity of Lie. Second, the transformation for which k = 1 is the 
correspondence 'of Segre, defined to be the correspondence between the ray- 
points corresponding to the point P x and the osculating planes at the point P x 
of all the curves on the surface S that pass through P x , as can be verified by 
eliminating \ X' homogeneously between equations (34) and (67) . 

To construct the point Pk corresponding to a given plane in any particular 
transformation of Cech with a definitely assigned value of k we may pro- 
ceed as follows. First draw the line conjugate to the line of intersection of 
the given plane and the tangent plane. On this conjugate line mark the 
point PQ which is the pole of the given plane with respect to the quadric of 
Lie, and the point PI which corresponds to the given plane in the corre- 
spondence of Segre. Then the required point P x can be located by making 
use of the fact that the cross ratio of the points P x , PQ, PI, Pk in the order 
named is k. This construction distinguishes the particular transformation 
of Cech from all the rest. 

* Lane, 1926. 3, p. 209. 



106 PROJECTIVE DIFFERENTIAL GEOMETRY 

24. Pangeodesics and the cone of Segre. The pangeodesics* on a surface 
are a covariant system of curves which will be precisely defined presently as 
extremals of a certain projectively invariant integral. These curves will 
also be characterized geometrically by a very simple property of their oscu- 
lating planes and their asymptotic osculating quadrics. The theory of the 
pangeodesics is closely connected with the theory of the plane sections of the 
surface and the dual curves, namely, the curves of contact of cones circum- 
scribing the surface. The envelope of the osculating planes, at a point P x of a 
surface S, of all the pangeodesics on S that pass through P x is called the cone 
of Segre of S at P x ; this cone was defined by Segre in a closely related way, 
which will be explained later on in this section. 

Let us give our attention now to defining the pangeodesics analytically. 
It is easy to verify that the differential expression ((3du*-\-ydv*)/dudv is 
absolutely invariant under the transformation (11), so that the integral 

(92) 

calculated along a curve on a surface S is invariant. Euler's equation (75) 
for the extremals of this integral reduces to 

(93) 2(/3+7X 3 )/X' = /U+2/U 2 -2 T uX 4 -7*X 5 

when v' is replaced by X. These extremals are called pangeodesics, and (93) 
is the differential equation of the pangeodesics on the surface S, in the sense 
that when X is a solution of this equation then v is determined as a function 
of u along a pangeodesic by means of equation (27) . It is perhaps worthy of 
note that equation (93) can be written in the form 



A very simple characteristic geometric property f of the pangeodesics 
can be discovered in the following way. The two asymptotic osculating 
quadrics (29), (31) at a point P x of a curve C x belonging to the family de- 
fined by equation (27) intersect in the asymptotic tangents through the 
point P x and in a conic which lies in a plane through P x . The equation of 
this plane is found, by eliminating x\ from equations (29) and (31), to be 



(94) { 5 



2X(/3+7X 3 ) fa- Xz 2 ) + [(/3+7X 3 ) (X' - /3+ T X 3 ) + (log 7 )u/3X 



* Fubini and Cech, 1926. 1, p. 141. 
t Lane, 1927. 3, p. 102. 



24] SURFACES IN ORDINARY SPACE 107 

If 0+7X 3 = 0, then C x is a curve of Darboux. Let us exclude this case. 
Demanding that the plane (94) coincide with the osculating plane (33) of the 
curve C x we find that X must be a solution of equation (93), and conversely. 
Thus we obtain the following characterization of the pangeodesics : 

A curve not a curve of Darboux on a surface is a pangeodesic if, and only 
if, at each of its points its osculating plane contains the conic of intersection of 
its asymptotic osculating quadrics that does not lie in the tangent plane of the 
surface at the point. 

It is possible to find another characterization of the pangeodesics. With 
this end in view let us seek for the differential equation of all plane sections 
of a surface S. A curve (7 X of the family defined by equation (27) on the 
surface S is a plane curve if, and only if, at every point x on the curve the 
four points x, x', x", x'" are co planar, accents indicating total differentia- 
tion with respect to u. Expressing these derivatives as linear combinations 
of x, x u , x v , x uv , and equating to zero the fourth-order determinant of the 
local coordinates of the four points, we obtain the differential equation of all 
the plane curves on the surface S, 



(95) 

where M is defined by 

(96) 

When X is replaced by dv/du equation (95) becomes an equation of the third 
order for v as a function of u along a plane curve. This fact is consistent with 
the fact that there are 3 planes in ordinary space. 

We interpolate here a few remarks on the relation of plane curves and 
union curves. Placing a = fr = in equations (84), one easily shows that 
the equation of the union curves of the projective normal congruence is M = 0. 
These curves are analogous to the geodesies of metric geometry, which can 
be defined as the union curves of the normal congruence (see 48, Chap. 
VI). Moreover, inspection of equations (60) and (95) shows that a union 
curve of the projective normal congruence is a plane curve if, and only if, it is a 
projective line of curvature. This is reminiscent of the metric theorem that a 
geodesic is a plane curve if, and only if, it is a line of curvature. This theo- 
rem, by a little calculation which will be omitted, can be extended to read: 

A union curve of any congruence Fi is a plane curve if, and only if, it is a IV 
curve of the congruence. 

Let us now turn to certain dual considerations. The curves of contact of 
cones circumscribed about a surface S mil be called cone curves. They are 
dual to the plane curves on S, in the sense that they correspond to the plane 



108 PROJECTIVE DIFFERENTIAL GEOMETRY 

curves in the dualistic transformation that converts every point P x of the 
surface S into the tangent plane of S at P x (see Ex. 13). Just as three con- 
secutive points of a curve C x on S determine an osculating plane of C x , so 
the tangent planes of the surface S at these three points intersect in the 
corresponding ray-point (66) of C x . The tangent planes at all points of C x 
envelop a developable surface whose edge of regression is the locus of the 
ray-point of C x . If this developable is a cone, then the ray-point is fixed as 
the point P x varies along the curve C x , and consequently the corresponding 
total derivative with respect to u of the expression (66) is proportional to 
the expression itself. From this proportionality we obtain the differential 
equation of all the cone curves on the surface $, 



(97) 

where N is defined by 

(98) 

The equation of the dual union curves of the reciprocal of the projective normal 
congruence is N = Q. 

The following statements concern relations among asymptotics, pan- 
geodesies, plane curves, and cone curves. If C x is both a plane curve and a 
cone curve then X must be a simultaneous solution of equations (95) and 
(97). Elimination of X" from these two equations leads to equation (93). 
Therefore, if a curve is both a plane curve and a cone curve, it is a pangeodesic. 
But it is easy to show that on an unspecialized surface S not every pangeo- 
desic is a plane curve. In fact, the asymptotics on S are pangeodesics, since 
X = is a solution of equation (93). Moreover, if an asymptotic curve is a 
plane curve then it is a straight line since, if X = is a solution of equation 
(95), then /3 = 0. Therefore a surface on which all pangeodesics, and hence 
the asymptotics, are plane curves is restricted to be a quadric. On a quadric 
surface every plane curve is also a cone curve, and vice versa, and the 
pangeodesics are indeterminate. 

If, for a particular pair of values of u, v which are the curvilinear co- 
ordinates of a point P x on a surface S, equation (95) is satisfied by a function 
X without the equation's being satisfied identically by X, then the curve C x 
of the family defined by equation (27) through the point P x has* at P x 
what is called a stationary osculating plane. At such a point the osculating 
plane of C x intersects C x in four consecutive points. A dual definition in- 
volving equation (97) can be given for a stationary ray-point of the curve C x , 

* Segre, 1908. 1, p. 409. 



25] SURFACES IN ORDINARY SPACE 109 

corresponding to P x . If the curve C x has at the point P x both a stationary 
osculating plane and a stationary ray-point then the positions of the osculat- 
ing plane and of the ray-point are restricted by the condition (93) so that, 
when the direction of the curve C x at the point P x is given, the osculating 
plane and the ray-point are determined. Those curves on the surface S whose 
osculating planes and ray-points are everywhere in these restricted positions are 
the pangeodesics. This is the second geometrical characterization of the 
pangeodesics. It may be restated as follows: 

The pangeodesics are those curves whose osculating planes at each point of 
the surface are precisely those planes which are able to serve as the osculating 
planes of curves each of which has at the point both a stationary osculating 
plane and a stationary ray -point. 

The stationary osculating planes and ray-points of curves at a point of a 
surface were investigated by Segre. He considered all curves on the surface 
through the point each of which has at the point both a stationary osculat- 
ing plane and a stationary ray-point, and.showed that the envelope of the 
planes is a cone of the sixth class, now commonly called the cone of Segre, 
while the locus of the ray-points is a curve of the sixth order, which is dual 
to the cone. It follows from the foregoing discussion that the envelope of the 
osculating planes at a point P x of all the pangeodesics through P x is the cone of 
Segre. Its equations are found by substituting the expression for X' given 
by equation (93) into equations (34) and then eliminating X homogeneously. 
The result is 



I -27^1+7^=^ = 0. 

The cone of Segre has its vertex at the point P x , and has the tangent plane 
of the surface S at P x for quintuple plane; in fact, it touches this plane along 
the asymptotic tangents and the tangents of Darboux through P x . Dually, 
the locus of the ray-points corresponding to the point P x of all the pangeodesics 
on the surface S through P x is Segre's curve of the sixth order in the tangent 
plane of S at P x . Its equations can be found directly, or else can be written 
immediately by means of the result of Exercise 13. 

25. The tetrahedron of Demoulin. Let us consider again the quadric of 
Lie at a point P x of a surface S. We propose to study the envelope of this 
quadric as P x varies on S. Clearly the surface S is part of the envelope, and 
it will be found that the quadric of Lie ordinarily touches its envelope, be- 
sides at P x , also in four other points. These four points are the vertices of a 
tetrahedron called the tetrahedron of Demoulin of the surface S, correspond- 
ing to the point P x . 



110 



PROJECTIVE DIFFERENTIAL GEOMETRY 



In order to find the points where the quadric of Lie touches its envelope, 
we shall differentiate the equation (26) of the quadric of Lie partially with 
respect to u and with respect to v, and then solve the two derived equations 
simultaneously with the original equation for the ratios of the local coordi- 
nates xi, . . . , 4 . But we observe that the tetrahedron of reference varies 
with the point P x , so that we need first of all to deduce some formulas for 
differentiating the local coordinates of a point referred to the local tetra- 
hedron Z, x U) x v , x uv at the point P x . We attack this problem immediately. 

The local coordinates of a point X near the point P x on the surface S are 
given by the power series (15). The local coordinates ?/, z, w of the corre- 
sponding points X u , X v , X uv , near the points x u , x v , x uv respectively, are 
represented to terms of the first order by the following power series: 



(100) 



w>s = irAtt+ 087+0) AH 
te?4=l+0Att+0 r AH ---- 



If a point has coordinates xi, . . . , x referred to the tetrahedron x, x u , x v , x uv 
at the point P x , and has coordinates Xi, . . . , X* referred to the neighbor- 
ing tetrahedron X, X u , X v , X uv , the equations of the transformation between 
the two tetrahedrons are found from the identity 



One substitutes for each of X, X u , X v , X uv the equivalent linear expression in 
x, x u , x v , x uv , and then equates corresponding coefficients of x, x u , x v , x uv . 
The resulting equations can be solved for Xi, . . . , X 4 by interchanging 
Xi, . . . , X 4 and x\, . . . , z 4 respectively and changing the signs of the in- 
crements AM, Av. The final result is 



(101) 



25] SURFACES IN ORDINARY SPACE 111 

Holding v = const, and taking the limit in these formulas as Au approaches 
zero (having first transposed certain terms and then divided by Aw), and 
then interchanging the roles of u and v and repeating the process, we obtain 
the desired formulas for differentiating the local point coordinates: 



(102) 



= 3 



If equation (26) is differentiated partially with respect to u by means of 
formulas (102), and if the derived equation is simplified by means of (26) 
itself and the second of the integrability conditions (10), the result can be 
written as the first of the following equations: 



(lOo) 



The second of these equations is the result of differentiating with respect to 
v and simplifying, and can be written easily by means of the substitution 
(30). Each of the two equations (103) represents a pair of planes. The 
equations of these planes can be written in the form 



(104) 



where /u, h 2 are the roots of the first of equations (103) regarded as a quadratic 
in XZ/XA', and similarly for fci, &2- Solution of each equation in the first row 
of (104) with each equation in the second row gives a contact point of the 
quadric of Lie with its envelope. Therefore, as the point P x varies, the quadric 
of Lie touches its envelope besides at P x also at the four points Pn, Pi 2 , ^22, Pn 
whose coordinates are 

(105) [hdi- (187+0/2, h>, /c y , 1] (i, j = 1, 2). 

These points are the vertices of a tetrahedron called the tetrahedron* of Demoulin 
at the point P x of the surface 8. 

The following properties of the tetrahedron of Demoulin will be stated 
here without proof. Figure 18 may assist in visualizing them. The four 

* Demoulin, 1908. 3, p. 494. 



112 



PROJECTIVE DIFFERENTIAL GEOMETRY 



lines PuPiz, PuPw, P&P21, PziPn are generators of the quadric of Lie forming 
a simple skew quadrangle, the first and third sides intersecting the v-tangent 
through the point P x and the other two sides intersecting the u-tangent. The 
points in which the lines P^Pzz and PziPn intersect the u-tangent are the 
flecnodes on this tangent regarded as a generator of the ruled surface of u-tan- 
gents circumscribing the surface S along the v-curve through the point P x ; and 




FIG. 18 

the lines PuPw, PziPn are the flecnode tangents at these points; similarly with 
u and v interchanged. The two lines PuPw, PwPzi are polar lines with respect 
to the quadric of Lie. The two planes PxPnPzz, PxPizPzi intersect in the first 
directrix d\ of Wilczynski. The four points Pij coincide in case the directrix 
curves (i.e., curves corresponding to the developables of the directrix con- 
gruences) coincide with the asymptotic curves. This subject will be taken up 
again in Section 41. 

Incidentally, we find by means of the formulas (102) the folio wing formu- 
las for differentiating local line coordinates: 



25] 



SURFACES IN ORDINARY SPACE 



113 



(106) 



0>13u = 012 1Ttti4 

Wl4w== 0)13 

0)23u = 0)13 0u 

0)421* = 0)14+0)23 20 U 0)42 



v ) 0)34 , 



0>12t; = 7^13 



0)12 V 0)14 



ut,) 0)42+ X0) 3 4 , 



0)42*= 0uO 

0)34u = 0)14+0)23 



EXERCISES 

1. A necessary and sufficient condition that a point whose coordinates are 
functions of u, v be fixed is that the coordinates satisfy two equations of the form 
Q, x v -\-qx = with p v = q u . 



2. By means of the transformation (3) with 21og\=0, reduce system (1) to 
Wilczynski's canonical form, 



Show that the transformation between Fubini's and Wilczynski's canonical forms 
is x = k($y) l ^y where k = const. 

WILCZYNSKI, 1907. 1, p. 246; FUBINI, 1918. 1, p. 1036 

3. Show that by a general transformation of parametric net system (1) takes the 
form 



Prove further that by a suitable choice of proportionality factor and of independent 
variables these equations can be reduced to the form x uu = Lx uv , x vv = Nx uv , for which 
the parametric curves are plane sections of the surface made by two axial pencils of 
planes with intersecting axes. 

4. If /3 = 7 = 0, the integral surfaces of system (1) are all quadrics, since the differ- 
ential equations can be reduced to the form x u u = 0, x vv = 0, of which four particular 
solutions are given by equations (II, 13). 



114 PROJECTIVE DIFFERENTIAL GEOMETRY 

5. Show by means of the series (18) that the integral surfaces of system (6) are 
surfaces of the third order in case 



4- W3 . 

LANE, 1927. 4, p. 475 

6. The quadric of Lie at a point P x of a surface S is the osculating quadric along 
the generator through P x of the ruled surface R u of asymptotic w-tangents construct- 
ed at the points of the v-curve through P x . Obtain the equation (26) of the quadric 
of Lie as the locus of the tangents of the curved asymptotics, (py-\-Ouv)dv -|-2dw = 0, 
of R u at the points x u -\-wx of the w-tangents of S through P x . 

7. The quadrics that are tangent to the quadric of Lie at every point of the two 
asymptotic tangents at a point of a surface are the quadrics of Darboux. 

8. If, at a point P x on a surface /S, the curve C x of the family defined by equation 
(27) has the tangent plane of S for stationary osculating plane, then C x is tangent 
to an asymptotic curve, and the asymptotic osculating quadric of C x which is de- 
termined by three consecutive asymptotic tangents of the other family is a quadric 
of Fubini, 

6(a^s-&ia*)-(207+3Ms4 = . 

FUBINI, 1928. 1, p. 16 

9. A curve C on a surface is a flecnode curve on the ruled surface of asymptotic 
w-tangents constructed at the points of C if, and only if, C is a curve of Darboux. In 
this case C is also a flecnode curve on the ruled surface of asymptotic y-tangents con- 
structed at the points of C. CECH, 1927. 7, p. 477 

10. At a point P x of a surface the harmonic conjugate of P x with respect to the 
two flecnodes on the generator xx u of the ruled surface R u (see Ex. 6) lies on the 
second directrix d^; similarly with u and v interchanged. 

FUBINI and CECH, 1926. 1, pp. 148 and 226 

11. The envelope of the rays, corresponding to a point P x of a surface, of all the 
conjugate nets of a pencil p\ is a conic (called the ray conic of p\ corresponding to 
P x ). Its equations are 0:4 = 4/370:2^3^ = 0, the definition of I being given in equation 
(71). This conic touches the asymptotic tangents through P x at the points where 
the flex-ray of the pencil p x crosses them. LANE, 1922. 3, p. 293 

12. At any point P x on a surface, the ray of any conjugate net N\h of a pencil p\ 
intersects the associate ray of N\h in a point Q on the flex-ray of the pencil p x . The 
flex-ray and the line joining P x to the point Q separate the ray and associate ray 
harmonically. 



SURFACES IN ORDINARY SPACE 115 

13. The dualistic correspondence which converts each point P x of a surface S 
into the tangent plane of S at P x is represented analytically by the transformation 



i = 4 , 2 = 3, 3= #2, 

accompanied by the substitution 

/ft 7 P q 

\-ft -7 ^r X 

LANE, 1926. 3, p. 206 



dv/du\ 
dv/du/ 



14. The locus of the axes, at a point P x of a surface, of all the conjugate nets of a 
pencil T?X is a quadric cone (called the axis quadric cone of the pencil p\ at the point 
Px). Its equations are i = 4/37 2 3 m 2 = 0, the definition of m being given in equa- 
tion (75). The tangent planes of this cone along the asymptotic tangents through 
P x intersect in the cusp-axis of the pencil p\. 

15. If the coordinates of the tangent plane at a point of a surface are defined 
by = e- (x, x u , x v ), and if x satisfies system (60, then the differential equations of 
the surface in plane coordinates are 



(0 = log fty) . 

16. The quadric (86) containing the asymptotic tangents at a point P x of a sur- 
face has third-order contact with the asymptotic curves at P x in case k\ 3. It 
has fourth-order contact with the curve C u in case ki= 3, & 2 = v/2, and has 
fourth-order contact with both asymptotic curves (7, C v in case k\= 3, k%= <f>/2, 
fc 3 = 1///2. Determine the quadric (86) that has sixth-order contact with C u . 

17. Consider a point P\ near a point P x on a surface S, and through Pi draw a 
line PiR meeting the quadric (86) in Oi, 2 , and meeting in T the tangent plane w of 
/S at P x . Let Oi be the intersection which approaches P x with Pi. Then as Pi ap- 
proaches P x the limit of the cross ratio (P\0\TR) is fcr, whatever be the curve 
along which Pi approaches P X) provided that it is not tangent to an asymptotic at 
P x ' f whatever be the line PiR, provided that it approaches a limit not in T; and what- 
ever be R, provided that it approaches a limit distinct from P x . 

BOMPIANI, 1924. 1, p. 417 

18. (The quadric of Moutard at a point P x of a surface S and in the direction of 
the tangent to the curve dv\du = Q through P x is defined to be the locus of the 
osculating conies at P x of the curves of intersection of S and the planes of a pencil 
with the tangent as axis.) Show by use of the series (18) that for the quadric of 
Moutard the coefficients in equation (86) have the values given by 

/c 3 =-(2 7 X 3 -/3)/3X 2 , 



116 PROJECTIVE DIFFERENTIAL GEOMETRY 

Two of the triple-point tangents of the curve of intersection of the surface S and the 
quadric of Moutard coincide, and the double tangent cuts the curve of intersection 
in five coincident points at the point P x . At a point P x on a surface S there are six 
tangents whose quadrics of Moutard pass through any given point not in the tan- 
gent plane of S at P x , and consequently there are at P x six plane sections of S whose 
osculating conies at P x pass through the given point. Through each tangent at P x 
there are two plane sections of S each of which has a six-point conic at P x . Through 
any line l\ at Px there are nine plane sections of S each of which has a six-point conic 
at P x . At P x there are ordinarily twenty-seven plane sections each of which has a 
seven-point conic at P x . At any point P x on a surface S the locus of the osculating 
conies of the plane curves of section of S and the planes through a fixed line li is an 
algebraic surface of the eighth order. Discuss this surface. 

MOUTARD, 1880. 3; oral communication to 
the Socie~te* Philomathique, 1865; letter to 
Poncelet, 1863. DARBOUX, 1880. 2, pp. 349 
and 363-72. WILCZYNSKI, 1909. 2, pp. 279 
and 288 

19. The correspondence between a variable point M in the tangent plane at a 
point P x of a surface and its polar plane with respect to the quadric of Moutard for 
the tangent conjugate to the line P X M is Segre's transformation. The correspond- 
ence between M and its polar plane with respect to the quadric of Moutard for the 
tangent P X M is the transformation of Cech for which k= 1/3. 

20. Show by the series (18) that all the < 4 non-composite cubic surfaces having 
fourth-order contact with a surface S at a point P x cut the tangent plane of $ at P x 
in the same nodal cubic curve, 



The line of inflexions of this curve is the second canonical edge ei of Green. 

B. SEGRE, 1927. 5, p. 729 

21. If a curve on an unspecialized surface in ordinary space is both a union curve 
for a congruence Ti and a dual union curve for the reciprocal congruence r 2 , then 
the curve is a curve of Segre and the congruences are somewhat restricted. More- 
over, there is a one-parameter family of Segre curves which have the designated 
property. If more than a single one-parameter family of curves have this property, 
then the congruences are the flex-ray and cusp-axis congruences of the Segre- 
Darboux pencil, and every Segre curve has the property. 

22. If the union curves of a congruence TI contain a conjugate net, then TI is 
the axis congruence of the net. Determine all the conjugate nets in the union curves 
of a congruence IV GREEN, 1919. 1, p. 133; BOMPIANI, 1924. 4, p. 11 

23. The three ray-points, corresponding to a point P x , of the three curves of Segre 
that pass through P x lie on the second axis a 2 of Cech. 



SURFACES IN ORDINARY SPACE 117 

24. At a point P x of a surface the osculating planes of any two Darboux curves 
and of the Segre curve conjugate to the third Darboux curve intersect in a straight 
line. BOMPIANI, 1924. 2, p. 51 

25. The ray-point cubic (70) and the ray conic (see Ex. 11) are tangent at three 
points on the tangents of Segre, the common tangents of the two curves being given 
by 



26. (The axis curves of a conjugate net are defined to be the Ti-curves of the axis 
congruence of the net, and the ray curves to be the T 2 -curves of the ray congruence of 
the net.) Show that the axis curves of a conjugate net N\ themselves form a con- 
jugate net in case X satisfies the equation 

2(log X),+03/X 2 ) tt -(7X 2 ) v = , 

and show also that the ray curves form a conjugate net in case X satisfies the same 
equation with the signs of and 7 changed. If both the axis curves and the ray 
curves form conjugate nets then (log X) U w = 0. (A conjugate net N\ with such a X 
is called isothermally conjugate.) For an isothermally conjugate net the cusp-axis 
curves (corresponding to the developables of the cusp-axis congruence of the pencil 
determined by the net) themselves form a conjugate net. By means of the trans- 
formation (11) obtain X= 1, so that the surface sustaining a conjugate net whose ray 
curves and axis curves both form conjugate nets is restricted by the condition M = 7. 

27. The extremals of the invariant integral f(p/v')du are hypergeodesics for 
which A =D = 0, B= (log 0)/2, C (log ), and for which the cusp-axis is a scroll 
directrix of Sullivan, namely, the line l\ for which a=^/2, 6 = ^/4. Similarly, the 
extremals of the invariant integral fyv' 2 du are hypergeodesics for which A =D = 0, 
B= (log 7) u , C= (log 7)0/2, and for which the cusp-axis is the other scroll direc- 
trix of Sullivan, nsbnely, the line l\ for which a = \f>/4, b = <p/2. 

SULLIVAN, 1915. 2, p. 202; FUBINI and CECH, 1927. 1, p. 684 

28. At a point P x on a surface the plane determined by the two scroll directrices 
of Sullivan (see Ex. 27) intersects the canonical plane in the canonical line for which 
k = 3/8; this is the line of intersection of the osculating planes of the three pangeo- 
desics tangent to the curves of Segre at P x * The canonical line for which k*= 3/4 
joins P x to the point of intersection, distinct from P x , of the quadrics of Moutard 
(see Ex. 18) for the three tangents of Segre, while the canonical line for which 
k= 5/12 joins P x to the residual point of intersection of the quadrics of Moutard 
for the three tangents of Darboux. 



29. If *>=^ = 0, then 0=t/F, y^h/UWV*, where h is an arbitrary constant 
and U is an arbitrary function of u alone, and V of v alone. By means of the trans- 
formation (11) obtain = 7 = 1 and by means of the integrability conditions (10) 
obtain q u = p v = Q, q v = p u , so that the equations (6) become 

(J, w, n=consts.) . 



118 PROJECTIVE DIFFERENTIAL GEOMETRY 

(An integral surface of these equations is called a coincidence surface because it has 
certain coincidence properties.) Show that at each of its points the canonical lines 
of the first kind coincide, and similarly those of the second kind also coincide, the 
canonical plane and canonical point being indeterminate. A surface is a coincidence 
surface if, and only if, on it the curves of Darboux are projective geodesies. The 
algebraic equation of a coincidence surface for which I = can be obtained by inte- 
grating the differential equations of the surface by means of elementary functions. 

WILCZYNSKI, 1913. 2; BOMPIANI, 1924. 2, p. 53 

30. Differentiating equation (51) with respect to u by means of formulas (106), 
prove that two consecutive osculating linear complexes at a point P x of a curve C u 
ordinarily intersect in the linear congruence oj 2 3 coi4=oj 3 4 = 0, which consists of the 
tangents of the surface R u (see Ex. 6) at the points of the w-tangent through the 
point P x . The curve C u belongs to a linear complex in case py (log 0) ttt , = 0. Simi- 
larly, Cv belongs to a linear complex in case 7 (log 7) Mt , = 0. If both C u and C v 
belong to linear complexes, then by the transformation (11) obtain = v, and by 
integrating the equation 2 -(log 0)^ = obtain 0=( U'V') 1 P/(U+V). Find p, q 
from the integrability conditions (10). 

31. Prove that any net N\ of curves on a surface S, such that at each point P x of 
S the two curves of N x form with the parametric asymptotic curves a constant cross 
ratio r, is represented by the equation (dv\du)(dvr\du) 0. Generalize the 
theory of conjugate nets, calling the class of all nets N\, for every A and a fixed r, a 
bundle B r , and rioting that the conjugate nets on S constitute the bundle B-\. 
Define a pencil p x of nets in the bundle B r so that any net N\h of the pencil is repre- 
sented by the equation (dv ^hdu)(dvr^hdu)=Q. Define the ray-point cubic of p^, 
and prove that it is independent of r. Prove that the envelope of the rays of all nets 
NX/I in the pencil p\ is a curve of class four and order six, with a double point at P x , 
the double-point tangents being the asymptotic tangents. This curve becomes the 
ray-conic, counted multiply, when l+r = 0. When l+r+r 2 = 0, and only then, the 
curve is tangent to the flex-ray of the ray-point cubic. Then the curve has the 
flex-ray for triple tangent, the contact points being where the Segre tangents meet 
the flex-ray. This curve has no inflexions, four double points, and six cusps. 

LANE, 1926. 4, p. 158 

32. (If the directrix curves on a surface form a conjugate net the surface is called 
isothermally-asymptotic.) Show that by a transformation of parameters it is possible 
to make = 7 in this case (see 51 and Ex. 20 of Chap. V). 

33. If a curve is such that its asymptotic osculating quadrics at each of its points 
intersect only in the asymptotic tangents at the point, the curve is a curve of 
Darboux. 

34. When the parametric curves dudv = on a surface S are any whatever, pro- 
vided that they do not form a conjugate net, any line l z in the tangent plane of S at a 
point Px can be defined as joining the points x u bx, x v ax. The tangent plane at 
the point z tt bx to the ruled surface of w-tangents constructed at the points of the 



SURFACES IN ORDINARY SPACE 119 

v-curve through P x is determined by the points z, x u , Xuvbx v . Similarly, the tan- 
gent plane at the point x v ax to the ruled surface of y-tangents at the points of the 
M-curve through P x is determined by x, x v , x uv ax u . The line h of intersection of these 
planes joins the points x and ax u bx v -\-Xuv- The two lines h, k, which are in the 
relation R, are in the reciprocal polar relation with respect to the quadric of Lie at 
the point P x of the surface S if the parametric curves are the asymptotic curves on S. 

GREEN, 1919. 1, p. 86; 1916. 3, p. 274 

35. (A surface is called a projectively minimal surface in case it is an extremal of 
the invariant integral J/07 dudv.) Such a surface is characterized by the property 
that on it and the four other sheets of the envelope of its quadrics of Lie the asymp- 
totic curves correspond. Another characteristic property of a projectively minimal 
surface is as follows : the four edges of the tetrahedron of Demoulin that lie on the 
quadric of Lie at a point of a surface S generate congruences whose developables 
correspond to conjugate nets on S if, and only if, S is a projectively minimal surface. 

THOMSEN, 1928. 4; 1921. 1, p. 233 and 6 

36. At a point P x of a surface S the quadrrc of Darboux 

xl = Q (k = const.) 



is the quadric of Lie if k 0, the canonical quadric of Wilczynski if k 1, and a quad- 
ric of Fubini (see Ex. 8) if A; =1/3. If any line li meets this quadric, besides at P x 
also at the point Pk, if l\ meets the quadric of Lie at the point P , and meets the 
quadric of Wilczynski at the point Pi, then 



so that by means of this cross ratio the quadric of Darboux is characterized geo- 
metrically for an arbitrary value of k. Use equations (102) to study the envelope of 
the quadric of Darboux with an arbitrary value of k as the point P x varies over the 
surface S. LANE, 1926. 3, p. 208; BOMPIANI, 1927. 11, p. 189 

37. The osculating plane (33) at a point P x of a curve C\ of the family dv\du = 
on a surface intersects the quadric (29) in a conic. The locus of this conic, as the 
curve C\ varies but remains tangent to a fixed line t at P x is the quadric 



When X-* oo this quadric approaches the canonical quadric of Wilczynski. When the 
quadric (31) is used in place of (29), the corresponding locus is the quadric 



which is the quadric of Wilczynski when X = 0. BOMPIANI, 1929. 2, p. 680 



120 PROJECTIVE DIFFERENTIAL GEOMETRY 

38. The two quadrics in Exercise 37 intersect, besides in the asymptotic tangents 
at the point P x , also in a conic which lies in the plane 



When the tangent t varies through P x , this plane envelops the cone (99) of Segre. 

39. Show that the constant c in the transformation (11) can be chosen so that the 
determinant (x, x u , x v , x uv ) has the value e. Placing X-XiX-\-x^u-\- x&v+x&w, 
prove that x\- (X, # w , x v , Xu V }e~ w , z 2 = (z, X, x V} x u v)e~ w , etc,, and then deduce the 
formulas (102) by direct differentiation. 

40. (A Roman surface of Steiner is a quartic surface each of whose tangent planes 
cuts it in two conies.) The parametric equations of the unique osculating Roman 
surface of Steiner, which has contact of the fourth order, at a point P x of a surface S 
can be written in the form 



34=24*1, . 

DARBOUX, 1880. 2, p. 359 



CHAPTER IV 
CONJUGATE NETS 

Introduction. The geometric theory of conjugate nets began with Dupin, 
but the analytic theory which commonly receives a geometrical interpreta- 
tion in connection with parametric conjugate nets is older and goes back to 
Laplace and Euler. If a complete list were made of all the mathematicians 
who have enriched the theory of conjugate nets by noteworthy contribu- 
tions, the list would include nearly, if not quite, all of the most distinguished 
geometers of the latter part of the nineteenth and the early part of the 
twentieth centuries. Some of these would be Koenigs, L. LeVy, Goursat, 
Darboux, Guichard, Ribaucour, Demoulin, Tzitz&ca, Bianchi, Segre, 
Bompiani, Wilczynski, Green, and Eisenhart. 

For the purposes of this and the following chapters it is expedient to 
revise our definition of a conjugate net. In Section 19 a net of curves on a 
surface in ordinary space was said to be a conjugate net in case at each 
point of the surface the two tangents of the curves of the net separate 
harmonically the asymptotic tangents of the surface, and it was proved in 
Section 21 that the tangents of the curves of one family of a conjugate net 
constructed at the points of each fixed curve of the other family form a de- 
velopable surface. But in space of more dimensions than three, a non- 
developable surface ordinarily has on it no asymptotic curve, and even if 
asymptotic curves exist on such a surface they do not form a net. Therefore 
we shall take the second property of a conjugate net just mentioned as our 
definition of a conjugate net in hyperspace. In ordinary space the two defi- 
nitions are equivalent, but in hyperspace the definition that we are now 
adopting is more fruitful. 

Whenever there is no danger of ambiguity a conjugate net may be called 
for brevity simply a net. In this sense a net is what is called in French un 
reseau. The net under consideration will be taken as parametric throughout 
most of this chapter. 

The contents of this chapter are disclosed by the following remarks. In 
presenting the theory of parametric conjugate nets it is natural, first of all, 
to introduce the equation of Laplace which is associated with a parametric 
conjugate net, and then to make inquiries as to what surfaces are capable 
of sustaining conjugate nets. The analytic theory of the transformation of 
Laplace, as interpreted by Darboux, leads to a sequence of conjugate nets 
determined by a given net and called a Laplace sequence. One interesting 

121 



122 PROJECTIVE DIFFERENTIAL GEOMETRY 

class of problems is concerned with the conditions under which such a se- 
quence can terminate. These and related matters occupy Sections 26-28. 
The theory of nets in the plane is taken up in Section 29, and conjugate 
nets in ordinary space are investigated in Sections 30-31. The conjugate 
and harmonic relations of nets and congruences are studied in Section 32, 
and the chapter closes with a brief account of the polar relation of two La- 
place sequences with respect to a hyperquadric. 

26. A surface referred to a conjugate net. The definition of a conjugate 
net which we are now employing may be stated as follows. A net of curves 
(as defined in Section 8) on a surface in a linear space of n dimensions is a 
conjugate net in case the tangents of the curves of one family of the net constructed 
at the points of each fixed curve of the other family form a developable surface. 
Although the two families do not enter this definition symmetrically, it will 
soon be shown that a conjugate net is symmetric in its two families. The 
purpose of this section is to introduce the Laplace equation that is associated 
with a parametric conjugate net, and to deduce a few of the simplest proper- 
ties of this equation. In particular, the Laplace-Darboux invariants H, K will 
be defined. 

Let us consider a surface S with the parametric vector equation x = x(u, v) 
in space S n . We shall now prove that a necessary and sufficient condition that 
the parametric curves on the surface Sform a conjugate net is that x satisfies an 
equation of Laplace. 

(1) x uv cx+ax u +bx v , 

where a, 6, c are scalar functions of u, v. Let us first suppose that the para- 
metric curves on S form a conjugate net. The point P v defined by 

y = x v + kx (k scalar) 

is on the ^-tangent at the point P x . As u varies, the point P y generates 
a curve whose tangent at P v is determined by P y and the point y u given by 

2/u = Xuv ] fcXu ~t~ KuX . 

This point is on the v-tangent in case y u is a linear combination of x and x vt 
s&yfx+gx v . Therefore, if the parametric curves form a conjugate net, then 
x satisfies the equation 



26] CONJUGATE NETS 123 

which is certainly an equation of Laplace. Conversely, if x satisfies equa- 
tion (1) and if P v is defined as above, then we find 

y u =(a+k)x u +bx v +(c+k u )x . 

The point y u is on the v-tangent if we take k = a. Therefore this tangent 
generates a developable surface when u varies. A similar argument can be 
made with the w-curves and v-curves interchanged. This completes the 
proof. The parametric conjugate net on the surface S can consistently be 
denoted by N x , since S is generated by a point P x . The symmetry of equa- 
tion (1) in u and v shows that the two families really play symmetrical 
roles in a conjugate net. 

From the foregoing demonstration it follows that the points Xi and x~\ 
defined by 

(2) Xi = x v ax , x~\ x u bx 

arc what would be called in the language of Section 21 the ray-points of the 
w-curve and the z;-curve respectively, corresponding to the point P x . These 
points are also called respectively the first and minus-first Laplace trans- 
forms (or transformed points) of the point P x with respect to the net N x . The 
Laplace transformation defined by equations (2) will be discussed more at 
length in Section 28. 

The effect of the transformation of proportionality factor, 

(3) ' x = \x (X scalar -^0) , 

on equation (1) is to produce another equation of the same form whose 
coefficients, indicated by dashes, can easily be shown by direct calculation 
to be given by 

f a = a-(logX) v , 5 = 6-(logX) w , 

(4) \ 

IXc = \ uv ~}-a\u-\-b\ v +c\ . 

This transformation leaves the net N x unchanged. It will be observed that 
if X is a solution of equation (1), then c = 0. Moreover, the net N x is also 
left unchanged by the transformation of parameters 

(5) w=/(w), v = V(v) (tf'WO) , 



124 PROJECTIVE DIFFERENTIAL GEOMETRY 

which converts equation (1) into another equation of the same form, whose 
coefficients, again indicated by dashes, are given by 

(6) a = a/F', b = b/U' , c = c/U'V . 

The Laplace-Darboux invariants H, K of the net N x are defined* by the 
formulas 

(7) # = c+a&-a M , K = c+ab-b v . 

These functions can easily be shown to be absolute invariants under the 
transformation (3), and are relative invariants under (5), being transformed 
by (5) according to the formulas 

(8) H = H/U'V , K^KJU'V . 

The invariants H, K are sometimes spoken of simply as the invariants of the 
net. 

If H = K then b v = a u and there exists a function X such that 

(log X) w = a , (log X) tt = b . 

If this function X is used in the transformation (3), then a 6 = 0, as is seen 
from equations (4). Therefore a Laplace equation with equal invariants can 
be reduced to the form 

(9) x uv = cx . 

27. Surfaces sustaining conjugate nets. There exist surfaces in hyper- 
space that do not have conjugate nets on them. The aim of this section is 
to show under just what conditions a surface may sustain one or more 
conjugate nets. 

In the definition stated in Section 10 for the space S(k, 0) at a point of a 
ruled surface the ruled surface can be replaced by any surface immersed in 
space $ n . In particular, the space S(2, 0) at a point P x of such a surface S is 
the ambient of the osculating plane at P x of every curve on S through P x . Since 
the osculating plane of a curve at the point P x is determined by the points 
x, x'j x", we see by letting X, X' vary in equations (III, 32) that the space 
S(2, 0) is ordinarily the space $5 determined by the six points x, x u , x v , x uu , x uv , 
x vv . However, in case x satisfies a differential equation of the form 

(10) Axuu+2Bx uv +Cx vv +Dx u +Ex v +Fx = Q , 
* Darboux, 1889. 1, Chap. II, p. 23. 



27] CONJUGATE NETS 125 

in which the six coefficients are scalar functions of u y v and not all of A, B, C 
are zero, the space S(2, 0) is at most a space S*. 

We shall now show that every integral surface of an equation of the form 

(10) sustains a conjugate net or else a one-parameter family of asymptotic 
curves. For this purpose let us consider in space S n a surface generated by a 
point P x with x satisfying equation (10) and let us carry out the transforma- 
tion of parameters 

(11) { = {(tt, v) , 7 7 = r;(w, tO (J = &,i7,-&, 
Differentiating x by the rules of elementary calculus we obtain 



(12) 



Substituting in equation (1) these expressions for the derivatives of x we 
get another equation of the same form as (10), the first three of whose 
coefficients, indicated by dashes, are given by 



(13) 



Incidentally it is- easy to verify that 
(14) AC-B* = 

and also that the determinant of the coefficients of A, B, C in the right mem- 
bers of equations (13) is J 3 . Consequently not all of A, B, C are zero. 

There are two cases to be discussed, according as AC IP^Q or AC B 2 
= 0. If AC- B 2 ? 0, the equation 



(15) 

has two functionally independent solutions, 



If these solutions are used as the functions , 17 in the transformation (11), 
then equations (13) show that .4 = = 0. Since 5^0, equation (10) is thus 



126 PROJECTIVE DIFFERENTIAL GEOMETRY 

transformed into an equation of Laplace with independent variables , rj. 
It follows that the ^-curves and rj-curves form a conjugate net. The parametric 
conjugate net ddr/ = associated with this equation of Laplace is the same 
net that in the original parameters u, v has the curvilinear differential 
equation 

(16) Cdu* - 2Bdudv+Adv 2 = , 

as can be verified by calculating the product ddr) from equations (11) and 
then making use of the well-known properties of the sum and product of the 
roots of the quadratic equation (15). 

If AC # 2 = 0, equation (15) has only one independent solution, 
0i = 0i (MI v). With this function 0i in place of and with r) = v in the trans- 
formation (11) we obtain A = 0, and therefore B = 0, C^O. It follows that 
x satisfies an equation of the form 

(17) x m +Dxt+Exi+Fx = Q . 

Consequently the rj-curves are asymptotic curves, since the osculating plane 
at every point of each r/-curve coincides with the tangent plane of the 
surface at the point. Thus the demonstration of the theorem is completed. 

Conversely, if a surface S immersed in space S n has on it a conjugate net, 
or else a one-parameter family of asymptotics, and if the parameters on S 
are suitably chosen, then x satisfies an equation of the form (1), or else 
of the form (17), each of which is of the form (10). Then the space S(2, 0) 
is at most a space $ 4 . We reach thus the following result : 

A surface with the parametric vector equation x x(u, v) immersed in space 
S n sustains a conjugate net or else a one-parameter family of asymptotic curves 
if, and only if, x satisfies an equation of the form (10); in geometrical language, 
if, and only if, the space S(2, 0) at each point of the surface is at most a space >S 4 . 

Contemplation of equation (10) leads to the following generalities. An 
unspedalized surface immersed in a space S n with n>4 does not sustain a con- 
jugate net, since the six or more homogeneous coordinates of a variable point 
on the surface cannot ordinarily be made to satisfy an equation of the form 
(10), whose coefficients yield only five essential ratios to be disposed of. 
However, the class of surfaces each of which does sustain a conjugate net in 
such a space is of considerable interest. On a non-developable surface im- 
mersed in space Si there is a unique conjugate net, or else a unique one-parame- 
ter family of asymptotics, since the five homogeneous coordinates of a vari- 
able point on the surface can be made to satisfy a unique equation (see 
Exs. 2, 3, 4) of the form (10). On a surface immersed in ordinary space >S 3 



28] CONJUGATE NETS 127 

there are infinitely many conjugate nets, since the four homogeneous coordi- 
nates of a variable point on the surface can be made to satisfy an equatior 
of the form (10) with one coefficient arbitrary. In fact, one of the families 
of a conjugate net can be assigned arbitrarily on such a surface; if the sur- 
face is not developable there are also two distinct one-parameter families 
of asymptotic curves on it ; if the surface is developable, the generators are 
the only asymptotic curves that the surface sustains. 

28. The transformation of Laplace. Equation (1) ordinarily admits o1 
two transformations into other equations of the same kind. These trans- 
formations are defined by equations (2), and are distinguished as being ir 
the positive or in the negative direction. Each of the new equations ordi 
narily admits of a similar transformation in one direction back into th< 
original equation and in the other direction into still another equation o 
the same kind, and so on. These transformations are called the transforma 
tions of Laplace because Laplace used them in investigating the possibility 
of integrating equation (1) by quadratures. The present section is devotee 
to a study of these transformations, which are referred to generically as th( 
transformation of Laplace. 

More specifically, the contents of this section may be summarized as 
follows. The first problem is to deduce formulas for the r'th and r'tl 
Laplace transforms of equation (1). This analytic theory received at th< 
hands of Darboux a geometrical interpretation in the theory of the so-callec 
Laplace sequence of conjugate nets determined by a given net. After intro 
ducing the Laplace sequence, some geometrical relations of the varioui 
nets of a sequence are studied, and the possibilities under which a Laplac< 
sequence can terminate are explained. 

The analytic theory of the Laplace transformation will now be briefly 
studied. Let us consider in space S n a surface S, with the parametric vecto 
equation x = x(u, v), referred to a conjugate net N x so that x satisfies equa 
tion (1). On the surface S let us consider a general point P x and the para 
metric curve C u through P x . The ray-point Xi of C tt corresponding to P x (o 
first Laplace transformed point of P x with respect to the net N x ) is define( 
by the first of equations (2). Differentiating x\ with respect to u and usinj 
equations (1), (7) we find 

(18) zi w = 6zi+#z. 

If H = 0, the point x\ is fixed when u varies, so that the locus of this poin 
as w, v vary is a v-curve. If #5^0, the locus of the point x\ is a surface 01 
which the parametric curves form a conjugate net Ni, since x\ satisfies * 



128 PROJECTIVE DIFFERENTIAL GEOMETRY 

Laplace equation, whose coefficients, invariants, and transforms, indicated 
by the appropriate subscripts, are given by 



(19) 



The last of these equations shows that just as the point x\ is the ray-point of 
the curve C u of the net N x , so the point x is the ray-point of the curve C v of the 
net Ni. 

It is customary to define a congruence of lines in hyper space to be < 2 lines 
which can be assembled into o l developable surfaces in two ways. Just as in 
ordinary space, the lines of such a congruence are the common tangents of 
two surfaces called focal surfaces of the congruence. The developables of 
the congruence determine on the focal surfaces two conjugate nets called 
the focal nets of the congruence. With these ideas in mind we see that the 
nets NX and N\ are the focal nets of the congruence of lines xx\ which are the 
v-tangents of N x and are at the same time the u-tangents of Ni. 

The foregoing considerations for the net N x can be repeated for the net Ni 
when HT*Q. The equation for the net Ni analogous to (18) for the net N x is 

(20) Xzu^iXi+H&i . 

If #i = 0, the locus of the point x% is a v-curve. If #i^0, the locus of this 
point is a net Nt, since x z satisfies a Laplace equation whose coefficients, 
invariants, and transforms, indicated by the appropriate subscripts, are 
given by 



(21) 



And so, by continuation of the argument, if HH\ Hr-z^O, we find 

(22) X r = Xr-l v a r -lX r -l , X ru = b r -.i 



28] 



CONJUGATE NETS 



129 



and if H r -i9*Q we obtain for x r a Laplace equation whose coefficients and 
invariants are given by 



a, = Or-l+ (lOg //,_!) , b r = 6 r _l = 6 , 

c r = c r _i+F r _i-K: r _i-6 r _i 
H, = 2ff r _, - K r ^ - (log #,_, 



(23) 



Similarly, interchanging u and f , a and 6, H and .K, and changing the signs 
of the subscripts, if KK-\ K- r +^Q, we find 



(24) 



#-r = X-r+1 u 



and, if K- r +i^Q, we obtain for X- T a Laplace equation whose coefficients 
and invariants are given by 



(25) 



a_ r = a_ r +i = a 



(r5D 



Let us observe more closely the geometrical aspects of the Laplace trans- 
formation. The sequence of nets 

. . . , N- r , . . . , AU ALi, N,, Ni, Ni,...,N r ,..., 

each of which can be derived from the two adjacent nets by transformations 
of Laplace, is called a sequence of Laplace and is indicated schematically in 
Figure 19. If // r = 0, the sequence terminates in the positive direction, 
since Nr+i reduces to a t>-curve, which is the locus of the vertices of cones 
circumscribing the net N r along the w-curves thereon. Similarly, if K- r = 0, 
the sequence terminates in the negative direction, since N- r -i reduces to a 
w-curve, which is the locus of the vertices of cones circumscribing the net 
N- r along the ^-curves thereon. It is natural to ask whether it is possible 
for the sequence to terminate under any other conditions than those just 
stated, which were first studied by Laplace* analytically. In order to 

* Laplace, 1893. 1, p. 5. Recherches sur le calcul integral aux differences partielles, 
"Mdmoires de TAcad&nie Royale des Sciences de Paris" (1773). 



130 



PROJECTIVE DIFFERENTIAL GEOMETRY 



answer this question let us suppose for simplicity that the locus Ni reduces 
to a curve; then Xi satisfies an equation of the form 



with not all of the coefficients A, B, C equal to zero. This equation is 
equivalent to 

Bx vv +(bA-aB+0)x 9 +[(H-ab)A-a v B--aC]x = Q . 

If BT*Q, then the surface S is ruled with its ^-curves for generators. In 
fact, S is developable and Ni reduces to a w-curve to which the generators 




FIG. 19 



of S are tangent (see Ex. 2). This case was first studied by Goursat.* If 
# = 0, since the surface S is non-degenerate it follows that 



Since not both of A and C are zero it follows that H = Q. This is the case of 
Laplace. It can happen that #5^0, H = Q. In this so-called mixed f case, 
the locus Ni reduces to a fixed point and the surface S is a cone with its 
vertex at this point, the generators of the cone being v-curves. We conclude 

* Goursat, 1896. 1. 

t Bompiani, 1912. 1, p. 393. 



28] CONJUGATE NETS 131 

that a sequence may terminate according to the case of Laplace, or according to 
the case of Goursat, or according to the mixed case. 

There are some interesting geometrical relations among the linear 
osculants constructed at corresponding points of curves of nets that belong 
to a sequence of Laplace. Equations (22), (24) show that the tangent line 
at a point of a v-curve of any net in a Laplace sequence is also the tangent line 
at the corresponding point of the u-curve of the adjacent net in the positive direc- 
tion of the sequence. It is not difficult to show that the osculating plane at a 
point of a v-curve of any net of a sequence is the tangent plane at the correspond- 
ing point of the surface sustaining the first transformed net, and is also the oscu- 
lating plane at the corresponding point of the u-curve of the second transformed 
net. For this purpose it is sufficient to demonstrate by simple calculations 
the truth of the equations 

(xi, xi u , Xi uu )=H 2 (x, x u , x v ) , (z_i, z_i v , x-i vv ) = K 2 (x, x u , x v ) , 

in which parentheses denote determinants. This amounts to proving that 
every linear combination of x\, x\ , xi uu can be expressed as a linear com- 
bination of x, x u , x v ; and that the same is true of every linear combination of 
x-ij x-i v , x-i vv . In a similar way it can be shown that the osculating space 
83 at a point of a v-curve of a net is also the osculating space $ 3 at the corre- 
sponding point of the w-curve of the third transformed net, being at the 
same time the space 83 determined by the tangent planes of the first and 
second transforms. Moreover, this space S 3 may be regarded as determined 
by the tangent plane and the osculating plane of the v-curve of the first 
transformed net or else by the tangent plane and the osculating plane of the 
u-curve of tlie second transformed net. In general, the osculating space S h at 
a point of a v-curve of a net is the osculating space & at the corresponding point 
of the u-curve of the kih transformed net (k>0). This space & may be re- 
garded as determined in various ways ; in particular, it can be regarded as 
determined by the tangent planes of the k 1 intervening transformed nets. 
We shall next discuss some geometrical properties of a sequence that 
terminates according to the case of Goursat, proving first that if the v- 
curves of a net in space S n are hyperplane curves, the sequence terminates in the 
positive direction according to the case of Goursat after nl transformations. 
If the ^-curves of a net N x in space S n are hyperplane curves, then each v- 
curve lies in a space $ n -i and therefore the coordinates x satisfy an equation 
of the form 



132 PROJECTIVE DIFFERENTIAL GEOMETRY 

which can be abbreviated into 2 Ux = 0. Since S Ux v it follows that S Uxi 
= 0. Moreover, since 



the accent indicating differentiation with respect to u, and since by equa- 
tion (18) we have SC7o;i M = 0, it follows that SC7'a;i = 0. Therefore each 
y-curve of the net Ni lies in a space $ n _ 2 , because x\ satisfies the two equa- 
tions 



Similarly, it can be shown that x?, satisfies the three equations 



so that each v-curve of the net N% lies in a space $ n -3. Continuation of the 
reasoning for a sufficient number of steps leads to the conclusion that the 
v-curves of the net N n -z are plane curves, that the 0-curves of the net N n -z 
are straight lines, and then that the ^-curves of the locus N n -i reduce to 
points, so that the net N n -2 lies on a developable surface with a u-curve JV_i 
for its edge of regression, and the sequence terminates according to the case 
of Goursat. The hyperplanes containing the ^-curves of the net N x are the 
osculating hyperplanes of the u-curve N n -i- Conversely, if the sequence 
terminates in the positive direction according to the case of Goursat, with 
the locus N n -i reducing to a u-curve, the osculating hyperplanes of this 
curve contain the ^-curves of the net N x . 

The geometrical situation when a sequence in space S n terminates ac- 
cording to the case of Laplace can be described briefly as follows. If the 
locus N n -i reduces to a v-curve, so that the Laplace sequence terminates in 
the positive direction according to the case of Laplace after nl transfor- 
mations, then # n _2 = and the second of equations (22) with r = n 1 gives 



By repeated use of the first of equations (22) with rn 1, ...,lwe find 
that x n -i can be expressed as a linear combination of z n _ 2 and its first 
^-derivative, or as a linear combination of # n _ 3 and its first two ^-derivatives, 
and so on until x n -\ is expressed as a linear combination of x and its first 
n~l ^-derivatives. Therefore, if the locus JV n -i reduces to a v-curve, then 
each point z n -i of this curve is the vertex of a cone circumscribing the 
net A^n-2 along a w-curve; the osculating planes of the y-curves at the points 
of the corresponding u-curve of the net AT n _ 3 pass through the same point 



29] CONJUGATE NETS 133 

z n _i, as do also the osculating spaces 3 of the v-curves at the points of the 
corresponding w-curve of the net N n _ 4 , and so on until finally the osculating 
hyperplanes of the v-curves at the points of the corresponding w-curve of 
the net N x pass through the same point z n _i. Conversely,* if the osculating 
hyperplanes of the v-curves at the points of each u-curve of a net N x pass through 
a point, then the sequence terminates in the positive direction according to the 
case of Laplace after n 1 transformations. 

29. Plane nets. After setting up the completely integrable system of 
partial differential equations that define a plane net except for a projective 
transformation, and writing the integrability conditions therefor, we shall 
limit the discussion in this section to two characterizations of plane nets 
with equal invariants, one of which is due to Koenigs and one to Green. 

The form of the differential equations defining a plane net can be dis- 
covered very quickly by the following geometrical considerations. Let the 
n+1 homogeneous coordinates of a point on a plane in space S n be given 
as analytic functions of two independent variables w, v, and suppose that 
the parametric curves form a net in the sense of Section 8 in the portion of 
the plane under consideration. Every such net in a plane is a conjugate net 
according to the definition of conjugate net given in Section 26, and there- 
fore x satisfies an equation of the form (1). Moreover, every curve in the 
plane is an asymptotic curve, according to the definition of asymptotic 
curve given in Section 8, so that x also satisfies a system of equations of the 
form (III, 1). Therefore then+l coordinates x of a variable point in a plane 
referred to any net whatever and situated in space S n are solutions of a system of 
equations of the form 



(26) 



.x vv =qx+yx u +dx v 



It may be well to remark that, since the point x is restricted to lie in a plane, 
at most three of the coordinates x are linearly independent, and for some 
purposes the remaining coordinates can be disregarded. We shall disregard 
them from now on in this section. This amounts to placing n 2. 

Of the four third derivatives of x there are two that can be calculated in 
two ways, since 



* Tzitz&ca, 1924. 3, p. 129; Bompiani, 1912. 1, p. 389. 



134 PROJECTIVE DIFFERENTIAL GEOMETRY 

Each of these equations reduces by means of system (26) to a linear first- 
order equation of the form (II, 2) all of whose coefficients must be zero. 
Thus we obtain six integrability conditions on the coefficients of system (26) : 



(27) 



Not only the original parametric net, but also any net into which it can 
be projected by a projective transformation of the plane, is an integral net 
of system (26). Conversely, the theory of linear partial differential equa- 
tions teaches us that this system of equations with the integrability condi- 
tions (27) satisfied defines a plane net except for such a projective trans- 
formation. 

We shall now prove a well-known theorem of Koenigs. * The projection of 
the asymptotic net on a surface in ordinary space from a point onto a plane not 
containing the point is a plane net with equal invariants. For this purpose let 
us consider in space $3 a surface whose parametric vector equation is x = 
x(u, v) referred to its asymptotic net. The four coordinates x satisfy equa- 
tions (III, 1) with $ a e = 0. Let us project the point P x from the point 
(0, 0, 0, 1) onto the plane z 4 = 0; the coordinates of the projection may be 
taken as the first three of the coordinates x. These three coordinates satisfy 
not only equations (III, 1) but also a Laplace equation and therefore satisfy 
a system of equations of the form (26). From the third and fourth of the 
integrability conditions (27) and from the definitions (7) we obtain 

(28) 8 u -a v = b v -a u = H-K . 

Therefore, for the plane net which is the perspective of the asymptotic net, 
we have H = K, as was to be proved. The reader may convince himself 
that there is no essential loss of generality in the choice we have made of 
the center and plane of projection. 

Green's geometric characterization f of a plane net with equal invariants 
has the advantage of depending on nothing outside of the plane of the net. 
He shows that ordinarily a given plane net determines uniquely a covariant 

* Koenigs, 1892. 1. t Green, 1918. 3. 



29] 



CONJUGATE NETS 



135 



net, called the congruentially associated net of the given net, with the property 
that, as a point varies along a curve of the associated net, the harmonic 
conjugate of the tangent of this curve with respect to the tangents of the 




FIG. 20 

given net at this point passes through the corresponding focal point of the 
ray of the given net. Before stating Green's characterization we shall find 
the curvilinear differential equation of the net congruentially associated 
with the parametric net of system (26). For this purpose let us consider a 
one-parameter family of curves dv \du-Q in the plane of Figure 20, 



136 P&OJECTIVE DIFFERENTIAL GEOMETRY 

and the curve C x of .this family through a piont P x . The ray of P x with 
respect to the parametric net joins the points xi, x-i defined by equations 
(2) ; consequently the point P v defined by 

y=x_i+kxi (fc scalar) 

is on the ray. As the point P x varies along the curve C x , the point P v gener- 
ates a curve whose tangent at P v is determined by P y and the point y r 
given by 



+[P+bk+k'+(d-a)\k]xi , 
where the functions P, Q are defined by 



This point is on the ray, so that P y is the focal point of the ray correspond- 
ing to the curve C x , if, and only if, the functions X and k are connected by 
the relation 

(29) P+K\+Hk+Q\k = Q . 

Moreover, the point P v lies on the harmonic conjugate of the tangent of 
the curve C x with respect to the parametric tangents, i.e., on the tangent of 
the curve C_ x at P x , in case k= X. Eliminating k from equation (29), and 
replacing X by dv/du, we obtain the desired differential equation of the net 
congruentially associated with the parametric net, namely, 

(30) Pdu? -(H- K)dudv - Qdv* = . 

Let us now denote by y\ the focal point of the ray that corresponds to the 
curve C x at the point P x . The other curve of the congruentially associated 
net through the point P x may be denoted by C M ; the focal point of the ray 
corresponding to C M , and lying on the tangent of the curve C_ M at P X) may 
be denoted by 3/2. Elimination of X, instead of k as in the preceding para- 
graph, leads to the equation 



The two foci y\, 7/2 are determined by solving this equation for fe, k% and 
substituting these two roots into the original expression for y. 



30] CONJUGATE NETS 137 

Green's characterization of plane nets with equal invariants is stated 
only for such nets as have proper congruentially associated nets. For the 
parametric net of system (26) this restriction obviously is (HK)*+ 
4PQ^O. The statement of the characterization is as follows. A plane net 
with a proper congruentially associated net has equal Laplace-Darboux in- 
variants (HK) if, and only if, at each of its points the tangents of the con- 
gruentially associated net separate the tangents of the given net harmonically. 
The demonstration is easily made by considering the harmonic, or simul- 
taneous, invariant of the binary quadratic form in equation (30) and of 
the form dudv. 

For a plane net with equal invariants the tangents of the curves C M and 
C_ x at the point P x coincide, as do also the tangents of C x and C_ M . Con- 
sequently the tangent of C M passes through the focal point y\ corresponding 
to C x , and the tangent of C x passes through the focal point y* corresponding 
to . Therefore a second form of Green's characterization may be stated 
as follows. 

A plane net with a proper congruentially associated net has equal invariants 
if, and only if, at each of its points the tangent of each curve of the congruentially 
associated net passes through that focal point of the ray of the given net that 
corresponds to the other curve of the congruentially associated net. 

30. Conjugate nets in ordinary space. This section and the next are de- 
voted to the theory of conjugate nets in ordinary space. The emphasis in 
the present section is placed on the fundamental and general aspects of the 
theory, while consideration of special classes of nets is for the most part re- 
served for the following section. The completely integrable system of differ- 
ential equations that define a conjugate net in ordinary space, except for a 
projective transformation, are written in a symmetrical form intimately 
connected with the axis congruence of the net. A canonical form of these 
equations suggests itself naturally and is used throughout the discussion. 
The developables and focal surfaces of the ray and axis congruences of a 
parametric conjugate net are determined in this section; the Weingarten 
invariants are introduced; and the point and tangential Laplace equations 
of a conjugate net are compared. 

The first problem is to establish the defining system of differential equa- 
tions. Let us consider in ordinary space $ 3 a surface S with the parametric 
vector equation x = x(u, v), referred to a conjugate net N x , so that x satis- 
fies equation (1). The osculating planes of the parametric curves C u , C v at a 
point P x on the surface $ intersect in the axis of P x with respect to the net 
N x . The axis does not lie in the tangent plane of S at P x . Let P v be any 



138 PROJECTIVE DIFFERENTIAL GEOMETRY 

point distinct from P x on the axis. Then x and y satisfy a system of differ- 
ential equations of the form 



x uv = ex +ax u +bx v , 



as may be seen by observing that the first and third equations express the 
fact that the point P v lies in the osculating planes of the curves C u and C v 
respectively at P x , while the second is merely the equation (1) of Laplace. 
We are going to choose for the point P v the harmonic conjugate of the point 
P x with respect to the two foci of the axis. In order to find these foci we ob- 
serve that the point P z defined by 

z y+kx (k scalar) 

is on the axis. When the point P x varies along a curve C x of the family 
dv \du = on the surface S, the point P z generates a curve whose tangent 
at Pz is determined by P e and the point z' given by 



In order to calculate expressions for the derivatives y u and y v as linear com- 
binations of x, x u , x v , y, it is sufficient to equate the two expressions for 
x vvu , and the two for x uuv , derivable from equations (31). Thus we find 

ry u =fx+mx u +sx v +Ay , 
(32) \ 

{yv = 



+nx v +By , 
where we have placed 



(33) 



Expressing z' as a linear combination of x, x u , x v , y, and equating to zero the 
coefficient of x tt , x v therein, we obtain conditions on the functions k, X 
necessary and sufficient that the axis may generate a developable and have 
P x for focal point when P x varies along C x , namely, 



= c v +ac+bq c8 q u , gL Cu+bc+ap COL p v 

= a v +a 2 ad q , tL = a u +ab+c a v , 

5 U , nL = b u +b 2 bap , 

) u , = a-(log L), . 



30] CONJUGATE NETS 139 

Elimination of X gives 

(34) k*+(m+ri)k+mn-st = Q . 

If fa, & 2 are the roots of this equation the corresponding points z\, 2 2 are the 
foci of the axis. The point y is the harmonic conjugate of the point x with 
respect to these foci in case m = n. We shall suppose from now on that 
m = n. Incidentally, on eliminating k and replacing X by dv/du, we ob- 
tain the differential equation of the axis curves of the net N x , in which the de- 
velopables of the axis congruence intersect the surface S, namely, 

(35) sdu 2 +2ndudv-tdv*=Q . 

The tangents of these curves are called the axis tangents of the net N x . At 
the close of this section we shall write equation (35) in another form. 

The integrability conditions of system (31) are found by equating the two 
expressions for y uv derivable from equations (32), and by reducing the result 
to a linear equation in x, x u , x v , y, all of whose coefficients must vanish. 
Thus one obtains 



(36) 



Certain consequences of these integrability conditions will be needed 
later on. The result of substituting the values of s, t from equations (33) 
into the last of these conditions is 

(37) (b+a+A) v = (a+d+B) u ; 

the result of substituting into this equation the values of A, B from equa- 
tions (33) is 

(38) 2(#-K)-(log r) MV = 5 tt -a v (r = AT/L) . 

The ratio r introduced here will occur again frequently. 

By means of equations (II, 9), (II, 10) the differential equation of the 
asymptotic curves on the surface S is easily shown to 

(39) 



s v n u +s8 2bn-\-f=sB nA , 



140 PROJECTIVE DIFFERENTIAL GEOMETRY 

In order to apply the same equations (II, 9), (II, 10) to the surface Si sus- 
taining the first Laplace transformed net JVi, we start from the first of 
equations (2) and find by direct calculations 

-bxi+Hx , 

(40) .~ 

= [(d-a)xi] v +2nNxi+tNx u +[nN v +N(n v +2an+g)]x 

+ (N v +BN)y. 

Indeed, the first of these equations is the same as equation (18), and the 
coefficients of the fourth are given in equations (19). The differential equa- 
tion of the asymptotic curves on the surface Si can now readily be shown to be 



(41) 

Similarly, or else by means of the substitution 

/u a c f p s n a A L H x r 
\vbcgqt-ndBNK X- 

the differential equation of the asymptotics on the surface _i, sustaining the 
Laplace transformed net JV_i, is found to be 

(43) sLdu*+Kdv 2 = Q . 

The asymptotic curves on the surfaces S and S-i correspond in case equa- 
tions (39) and (43) are equivalent. Then the w-tangents of the net N x form 
a congruence of the special type called a W congruence. Thus we reach the 
result that the u-tangents of the net N x form a W congruence in case the Wein- 
garten invariant W^ u \ defined by 

(44) W^ = sN-K, 

vanishes. Similarly, the v-tangents of the net N x form a W congruence in case 
the Weingarten invariant W (v \ defined by 

(45) WM = tL-H , 

vanishes. Substituting in equation (38) the values of 8 U , a v given by equa- 
tions (33), then eliminating s, t by means of the definitions (44), (45), and 
employing finally the definitions (7), we arrive at an important relation, 

(46) WM - WW = (log r) uw (r = N/L) . 



30] CONJUGATE NETS 141 

We shall next determine the developabks and focal surfaces of the ray con- 
gruence of the net N x . The point Pj- defined by 

f = x-i+ kxi (k scalar) 

is on the ray of N x corresponding to the point P x . When P x varies along a 
curve C x of the family dv \du=Q on the surface S, the point Pf generates 
a curve whose tangent at Pj- is determined by Pj- and the point f ' given by 



By use of the first two of equations (40) and the symmetric equations writ- 
ten by means of the substitution (42), the derivative f ' can be expressed as 
a linear combination of z_i, #1, x, y. Equating to zero the coefficients of x, y 
in this expression we obtain conditions on the functions fc, X necessary and 
sufficient that the ray may generate a. developable and have Pf for focal 
point when P x varies along C x , namely, 

-nL+K\+Hk+nNk\ = Q , L+Nk\ = Q . 
Elimination of X gives 

(47) NHk* - 2nLNk - LK = . 

If ki, k 2 are the roots of this equation, then the corresponding points fi, f 2 
are the foci of the ray. Moreover, elimination of k gives, on replacing X by 
dv/du, the differential equation of the ray curves of the net N x , which correspond 
on the surface S to the developables of the ray congruence, namely, 

(48) LHdu*+2nLNdudv-NKdv 2 = Q . 

The tangents of these curves are called the ray tangents of the net N x . 

Another method of determining the foci of the ray yields also an interest- 
ing theorem. We find, by differentiation and substitution, 

( u =(a-b)x-i+(k u +bk)xi+(Hk-nL)x+Ly , 
I f 9 =ax-i+[k v +(d-a)k]xi+(K+nNk)x+Nky . 

Let us consider any plane through the ray (containing therefore the points 
z_i, Xi) and meeting the axis in a point y+px. We wish this plane to be what 
is called a focal plane of the ray; so we impose the condition that it be 



142 PROJECTIVE DIFFERENTIAL GEOMETRY 

tangent at a point f to a focal surface of the ray congruence. This condition 
is simply that the plane contain the points f u , f r ; thus we find 



It follows that k must be a root of equation (47) ; in this way we have deter- 
mined the foci of the ray again. If MI, M2 correspond respectively to the roots 
ki, & 2 of equation (47), a simple calculation gives Mi+M2 = . Thus we have 
proved* the following theorem: 

At a point P x of a surface sustaining a conjugate net N x in ordinary space 
the harmonic conjugate of the tangent plane of the surface, with respect to the 
two focal planes of the ray of N x , passes through the point y which is the har- 
monic conjugate of P x with respect to the two foci of the axis of N x . 

Equation (1) is sometimes called the point equation of the net N x . The 
coordinates of the tangent plane of the surface S sustaining N x also satisfy 
an equation of Laplace which is called the tangential equation of N x (see 
Ex. 19). In order to find this equation in a simple form, we observe that 
equation (37) shows that there exists a function 6 defined, except for an 
additive constant, by the differential equations 

B u = b+a+A , d v = a+d+B . 

If the coordinates of the tangent plane at a point P x of the surface S are 
defined by the formula 

(50) S = <T 9 (x,x u ,x v ), 

then direct calculation shows that satisfies the Laplace equation 

(51) = (tL-AB-AJS-B^-AS, . 

This is the tangential equation of the net N x . 

Sometimes the Laplace-Darboux invariants H, K of equation (1) are 
called point invariants of the net N x , while the corresponding invariants 
J^f J^of equation (51) are called tangential invariants of N x . These invari- 
ants and the Weingarten invariants defined by equations (44), (45) are 
connected by the relations 

(52) Jti 
*Slotnick, 1931. 1, p. 148. 



31] CONJUGATE NETS 143 

Consequently the equation (35) of the axis curves can now be rewritten in 
the form 



Moreover, the equations (41), (43) of the asymptotic curves on the surfaces 
Si, S_i become respectively 

LHdu?+NJfdv z = , L jSfdtf+NKdv* = . 

31. Special classes of conjugate nets in ordinary space. There are cer- 
tain classes of conjugate nets distinguished by possessing special properties. 
One of the most interesting of these properties is that of isothermal con- 
jugacy, and so the class of isothermally conjugate nets will be the first to 
claim our attention in this section. Two other important classes are the 
class of harmonic nets and the class of R nets. These and other classes will 
be discussed, and incidentally the theory of an unspecialized net will be 
amplified to some extent, for example, by the calculation of the differential 
equations of the surface sustaining the first Laplace transformed net. 

The property of isothermal conjugacy of a net of curves on a surface in 
ordinary space was first defined* by Bianchi. His definition was of an ana- 
lytic nature; although it differs in form from the definition stated in Exer- 
cise 26 of Chapter III, the two definitions are in fact equivalent. Bianchi 
showed that the property of isothermal conjugacy is of a projective char- 
acter, but he did not give a purely geometrical description of it. No serious 
attempt seems to have been made to discover such a description until 1915 
whenf Wilczynski found an algebraic relation between certain geometrically 
interpreted projective invariants which is characteristic of isothermally 
conjugate nets. In 1916 Green gavel the purely geometric description 
embodied in Exercise 8, which is now known to be characteristic only of 
non-harmonic isothermally conjugate nets, although Green mistakenly 
supposed at first that he had completely solved the problem of characteriz- 
ing isothermally conjugate nets geometrically. Wilczynski was the first to 
reach a complete solution! of the problem. His final solution is embodied 
in Exercise 26 of Chapter III, and was given|| its present analytically simple 
form by the author. 

We now undertake to find an analytic condition that the parametric 
conjugate net N x of the present chapter be isothermally conjugate. For this 
purpose we make a transformation of curvilinear coordinates from the 

* Bianchi, 1922. 5, p. 210. 

t Wilczynski, 1915. 1, p. 323. Wilczynski, 1920. 1, p. 221. 

t Green, 1916. 1, p. 313. || Lane, 1922. 3, p. 292. 



144 PROJECTIVE DIFFERENTIAL GEOMETRY 

asymptotic coordinates u, v of Chapter III to the conjugate coordinates of 
the present chapter, which will be denoted in just this paragraph by , 77. 
Such a transformation is represented by 

* = {(*, v) , II = I?(M, v) 
and differentiation gives 
(53) d = 



Now the net $#77 = is to be the same as the net efo 2 X 2 du 2 = of Chapter 
III; it follows that 

fcifc+fcifc = , &,i7u+X 2 &ifc = . 

These conditions can be satisfied by taking , rj as solutions of the equa- 
tions 

(54) u=-X v , i? = Xi7, , 

from which one obtains, by differentiation and substitution, 

v , (X w XX v )t, 

(55) 



Solution of equations (53) for du, dv, elimination of , rj u by (54), and sub- 
sequent multiplication result in 



Therefore the differential equation of the asymptotic curves in conjugate 
parameters is 



(56) ijMf 

Comparison of this equation with equation (39) shows that 



As in calculating equations (12) we find, on differentiating any function 
x of u, v, 



X v 



31] CONJUGATE NETS 145 

Solution of these two equations for x$ and z, furnishes two differentiation 
formulas, 



(57) a? * - (x v -\ 

by the aid of which we obtain by direct calculation* of the derivatives of r, 

(58) Xfci? 



In Exercise 26 of Chapter III the defining condition for isothermal con- 
jugacy was (log X) ttt , = 0. Our result may therefore be stated as follows: 
A parametric conjugate net d!-dri = Qis isothermally conjugate if, and only if, 



The form of system (31) is left unchanged by every transformation of the 
group 

(59) x = \(u, v)x , y = n(u, v)y , u= U(u) , v = V(v) 



The effect of such a transformation on the coefficients of system (31) is easily 
calculated (see Ex. 25). In particular, we find that the effect of the trans- 
formation x = Xz on the coefficients a, 8 is given by 

(60) a = a-2(logX) w , 6 = 5-2 (log X), , 

but we shall not need to write the formulas for the other coefficients. Simi- 
larly, the effect of the transformation u=U(u), v*= V(v) on the coefficients 
L, N is found to be given by 

(61) L = L/U' 2 , JV=AT/F' 2 . 

If the parametric net dudv Q is isothermally conjugate, then (log r) tt t> = 
and hence r= U\(u)Vi(v) where U\, V t are arbitrary functions of the argu- 
ments indicated. It follows from equations (61) and the definition of r in 
(38) that it is possible to make a transformation of parameters so that after 
the transformation we shall have r=l. In fact, it is sufficient to use 
functions U, V satisfying the conditions U' 2 Ui= -V'*/Vi = const. Thus 
we reach the conclusion: 

It is no restriction on an isothermally conjugate net to suppose that the pa- 
rameters have been chosen so that r = 1. 

Another special class of conjugate nets consists of harmonic nets. A con- 
jugate net for which n = Qis called} a harmonic net because it possesses a num- 

* Ibid., p. 288. t Wilczynski, 1920. 1, p. 215. 



146 PROJECTIVE DIFFERENTIAL GEOMETRY 

ber of harmonic properties. Some of these properties will now be indicated, 
under the supposition that the harmonic conjugate net considered is not 
further specialized. Equation (35) shows that at each point of a harmonic 
net the axis tangents separate the tangents of the net harmonically. Moreover, 
since the axis tangents and the axis determine the focal planes of the axis, it 
follows that at each point of a harmonic net the focal planes of the axis sepa- 
rate the osculating planes of the net harmonically. Equation (48) shows that 
at each point of a harmonic net the ray tangents separate the tangents of the net 
harmonically, while equation (47) shows that the foci of the ray separate the 
ray points harmonically. Finally, the second of equations (40) and the sym- 
metric equation 'that can be written by means of the substitution (42) show 
that at each point P x of a harmonic net N x the point y, which is the harmonic 
conjugate of P x with respect to the foci of the axis, is also the point of intersec- 
tion of the v-tangent at the corresponding point x : of the net Ni (which joins 
the points x\, Xz) and the u-tangent at the corresponding point x~\ of the net 
N-i (which joins the points z_i, 3-2). 

It is perhaps worthy of remark that two corresponding points P x , P y 
for an unspecialized conjugate net N x are not only separated harmonically by 
foci y(n?+J%y/LN) l/2 x of the axis of P x , but also by the points ynx 
where the ^-tangent of the net Ni and the u-tangent of the net JV_i meet the 
axis, as well as by the points y (n 2 +HK/LN) l/2 x where the focal planes of 
the corresponding ray intersect the axis. The point y-\-nx could also be de- 
fined as the point where the osculating plane of the #-curve of the net N x 
is tangent to the ruled surface of axes constructed at the points of the in- 
curve through the point P x , and there is a symmetric definition for the 
point y nx. 

A third special class of conjugate nets consists of R nets. A conjugate net 
for which W^ = W ( ^ = Q is called* an R net; each family of curves of an R 
net have tangents that form a W congruence. Inspection of equation (46) 
shows that an R net is isothermally conjugate, and that an isothermally con- 
jugate net such that the tangents of the curves of one family form a W congruence 
is an R net. 

Some of the facts concerning ray curves and axis curves that form conjugate 
nets are as follows. Calculation of the harmonic invariant of the binary 
quadratic forms in equations (39) and (48) shows that the ray curves of a 
conjugate net themselves form a conjugate net\ if, and only if, H = K. Simi- 
larly, the axis curves form a conjugate net in case ffi= J#C Furthermore, if the 
ray and axis curves of a conjugate net both form conjugate nets, the fundamental 

*Tzitz&ca, 1911. 1. 

t Wilczynski, 1915. 1, p. 319. 



31] CONJUGATE NETS 147 

net is isothermally conjugate. Moreover, an isothermally conjugate net has 
ray curves forming a conjugate net if, and only if, its axis curves also form a 
conjugate net. For such a net we find from equation (38) that 5 W a = 0; 
when this condition is satisfied, equations (60) show that by a transforma- 
tion of proportionality factor it is possible to make a = 5 = 0. 

Investigation of the possibilities as to coincidence of the ray curves and 
axis curves yields the following results. The ray curves and axis curves of a 
non-harmonic conjugate net coincide in case H = 3tf?K = JtfC If, in addition, 
the fundamental net is isothermally conjugate, then the ray-and-axis curves 
form a conjugate net, and the fundamental net is an R net. Furthermore, 
if a non-harmonic conjugate net has HK^Q, the ray curves and axis curves co- 
incide in case the developables of the ray congruence intersect the surfaces Si, S-i 
in conjugate nets. The ray curves and axis curves of a harmonic net coincide in 
case HJfc- Kffl= 0. If, in addition, the fundamental net is isothermally con- 
jugate, then either the ray-and-axis curves form a conjugate net or else the 
fundamental conjugate net is restricted by the conditions 



The following statements concerning the ray tangents and axis tangents 
are of some interest. Equations (47), (48) show that the ray tangents at 
each point of a non-harmonic conjugate net meet the corresponding ray in the 
focal points of the ray if, and only if, the ray curves form a conjugate net. The 
ray tangents at each point of a harmonic net meet the corresponding ray 
in the focal points of the ray under either of two hypotheses; first, if the 
ray curves form a conjugate net ; second, if the fundamental net is restricted 
by the condition H-\-K = Q, i.e., if the ray curves coincide with the asymp- 
totic curves. The axis tangents at each point of a non-harmonic net meet the 
corresponding ray in the focal points of the ray, if, and only if, the net is an R 
net. The axis tangents at each point of a harmonic net meet the correspond- 
ing ray in the focal points of the ray in case the fundamental net is re- 
stricted by the condition HJ^== KJtfC 

Referring to Figure 21, we propose to find a system of equations of the 
form (31) for the first Laplace transformed surface Si of an integral surface 
S of system (31). The second equation of the system for the surface Si, 
namely, the Laplace equation, was found in Section 28, its coefficients being 
given in equations (19). In order to find the other two equations let us ob- 
serve that the point X defined by 

(62) X = x u +[A-a-(\ogO]x 

is certainly on the u-tangent at the point P x of the parametric net N x on 
the surface S. Moreover, the point X is also on the axis of the point Xi 



148 PROJECTIVE DIFFERENTIAL GEOMETRY 

with respect to the net Ni on the surface &, since the point X can be shown 
by equations (40) and the first of the integrability conditions (36) to lie in 



axis of 




FIG. 21 

the osculating planes at the point Xi of the curves C u and C v of the net 
Furthermore, the point 



(63) 



31] CONJUGATE NETS 149 

where n\ is defined by placing 

(64) 2r e ^n 1 =(5-o-ai) v +(5-a~ai)(log H/r) v +2nN , 



is certainly on the axis of the point Xi, and can be shown by means of equa- 
tions (40) and the first of the integrability conditions (36) to be also on the 
0-tangent of the net N* In fact, the point (63) is identical with the point 



d-a- (log H/r) v ]x 2 . 
The harmonic conjugate y\ of x\ with respect to X and X -\-Zn\Xi is given by 
(65) yi 



That the function n\ is, as the notation indicates, the coefficient n for the 
net Ni is verified by observing that the points X and X -\-2n\xi can be 
expressed in the respective forms 

(66) yi-n&i , yi+nixi . 

Now that we know the expression for y\ the rest of the calculation for the 
required system of equations for the surface Si is largely mechanical. Use 
is made of equations (33), (40), (62), (65). The result is a system of equa- 
tions of the form (31) whose coefficients, indicated by the appropriate sub- 
script, are given by the following formulas: 



i = a+(log 



(67) 



As an application of these formulas, let us calculate the Weingarten in- 
variants W[ u \ W( v) and the invariant r\ for the net Ni. We find 

(68) Fi tt) = WM , W ^ = W<* - (log n) ttv , n = r Jf/H . 

It is easy to see that, if TF<"> = W = 0, then W[ u) = W[ v} = 0. Thus we 
prove the theorem : 

A Laplace transform of an R net is an R net. 



150 PROJECTIVE DIFFERENTIAL GEOMETRY 

32. Conjugate and harmonic relations of nets and congruences. When 
the points of a conjugate net and the lines of a congruence in space S n are in 
one-to-one correspondence, there are two special relative positions of the 
net and congruence which are of particular interest. In one of these posi- 
tions the net and congruence are said* to be conjugate to each other, and in 
the second harmonic to each other. These relations of net and congruence 
will be explained more precisely presently, and then it may be observed 
that they are dual relations in the dualistic transformation that converts 
each point of a net into the tangent plane at this point of the surface sus- 
taining the net, which may be called simply the tangent plane of the net. 

Various properties of the conjugate and harmonic relations of nets and 
congruences in space S n will be studied in this section. The problems of de- 
termining analytically all nets conjugate, or harmonic, to a given con- 
gruence will be solved, as well as the problems of determining all congru- 
ences conjugate, or harmonic, to a given net. The correspondence between 
two nets known as the Levy transformation will be introduced. 

The conjugate relation of a net and a congruence is defined as follows. 
A net and a congruence, such that there is just one line of the congruence through 
each point of the net and not in the tangent plane of the net at the point, are con- 
jugate to each other in case the developables of the congruence intersect the sur- 
face sustaining the net in the curves of the net. A focal net of a congruence 
cannot be conjugate to the congruence. A bundle of lines, such that there 
is just one of its lines through each point of a net and not in the tangent 
plane at the point, is said to be conjugate to the net. 

We shall now determine all nets that are conjugate to a given congruence. 
Any congruence IV with non-degenerate focal nets Nr N{ in space S n can 



FIG. 22 

be represented analytically by writing the equations which assert that the 
u-tangent at a point 77 is also the v-tangent at a point f , as in Figure 22. 
These equations take the form 

(69) T/ = ar/+0i*, f = 7>7+5f (/fy^O) . 

Indeed, by a transformation r? = \rj, = /*? it is possible to reduce each of a, 6 
to zero, but we shall not make use of this simplification for the present. 
* Guichard, 1897. 2, pp. 478 and 483. 



32] CONJUGATE NETS 151 

Elimination of f from equations (69) furnishes the Laplace equation for 
the net AT,, namely, 

(70) r) uv = [0y-ad+a v -a(\og 0),b+[+(log ftj^ + oiy, , 

while elimination of rj would result in the Laplace equation for Nf. Any 
point P x on the line l^ can be defined by placing 

(71) X = M Xf (X, \t, scalars) . 
If the functions X, M satisfy equations (69), i.e., if 

(72) X tt = aX+ftz , /xv = 7X+ M , 

it can be shown that the point P x generates a net N x conjugate to the con- 
gruence IV. In fact, differentiation of equation (71), followed by judicious 
use of (69), (72), leads to 

(73) x u ax = nu'n Xf tt , x v 8x = w v X r f . 

Differentiation of the second of these equations with respect to u, and sub- 
sequent use of equations (69), . . . , (73), lead finally to a Laplace equation 

for the net N x , 



X) v (log /OJs 
+ [*+(log X)Jx+[a+(log M)J*. . 

Conversely, If the point P x defined by equation (71) generates a net con- 
jugate to the congruence IV, then the functions X, ^ can be multiplied 
by such a common factor, without changing P x , that they satisfy equations 
(72). The proof* will be omitted. Thus we arrive at the conclusion: 

A congruence IV represented by equations (69) is conjugate to every net N x 
defined by equation (71) subject to the conditions (72), and every net N x con- 
jugate to the congruence IV can be so represented. 

The harmonic relation of a net and a congruence is defined as follows. A 
net and a congruence, such that there is just one line of the congruence in each 
tangent plane of the net and not through the point of contact, are harmonic to 
each other in case the developabks of the congruence correspond to the curves of 
the net. A focal net of a congruence cannot be harmonic to the congruence. 
Just as the focal planes of a generator of a congruence conjugate to a net 
contain the tangents of the net at the point on the generator, so the focal 

* Tzitz&ca, 1924. 3, p. 69. 



152 PROJECTIVE DIFFERENTIAL GEOMETRY 

points of a generator of a congruence harmonic to a net lie on the tangents 
of the net that are in the tangent plane containing the generator. 

The problem of determining all congruences that are harmonic to a given net 
can be solved in the following way. Let us consider a net N x for which 
equation (1) is valid. The points P p , P ff defined by 

(75) p = # M + hx , (r = x v +kx (h, k scalars) 




FIG. 23 



are on the tangents of the curves C UJ C v respectively at the point P x as 
indicated in Figure 23. Simple calculation leads to 

f p v = ap+(b+h)<T+(h v +c ah bk hk)x , 
\a u =(a+k)p+b<r-\-(k u +c ah bk hk)x . 

If the line l ptr generates a developable surface as u varies, with P as focal 
point, the coefficient of x in the second of equations (76) must vanish; 
similarly, if l pff generates a developable as v varies, with P p as focal point, 
the coefficient of x in the first of (76) must vanish. Therefore every congru- 
ence harmonic to a net N x with x satisfying equation (I) can be generated by 
the line l p<r joining the points P P) P 9 defined by equations (75) with the functions 
h, k satisfying the conditions 

(77) h v = k u = ah+bk+hk-c . 



32] CONJUGATE NETS 153 

These points P and P p will be called respectively the first and second focal 
points of the line l ptr . 

It is easy to show that x can be multiplied by such a factor that any con- 
gruence harmonic to the net N x will be generated by the line joining the points 
x u , x v . This amounts to showing that by a transformation z = X it is pos- 
sible to reduce both of h, k to zero. We find 



It is sufficient for our purpose to choose the function X as a solution of 
(log X) w +/i = , (log X) v +fc = , 

and this choice is possible because h v = k u . After this simplification has been 
made, equation (77) reduces to c = 0. This is, in a sense, the simplest ana- 
lytic representation of a congruence haxmonic to a net. 

The transformation known as the Levy transformation is defined as fol- 
lows. A Levy transformation between two nets is the correspondence between a 
focal net of a congruence and any net conjugate to the congruence. Formulas 
for Levy transforms* of a net N x with x satisfying equation (1) will now be 
deduced. The equations of the form (69) for the congruence of w-tangents 
of the net N x are 

(78) x u = bx+x-i , x-i v = Kx+ax-i , 

and so the most general net conjugate to this congruence is generated by the 
point p defined by 

(79) p = 0_iz-0z_i , 
where the functions 0, 0_i satisfy 

(80) = 



The function is easily shown to be a solution of equation (1), and the nota- 
tion 0_i is intended to be suggestive. Therefore any Levy transformed net N p 
of the net N x along the congruence of u-tangents of N x is generated by the point 
p given by 

(81) p = 0uX-0Ju, 

* Levy, 1886. 2, p. 67. 



154 PROJECTIVE DIFFERENTIAL GEOMETRY 

where 6 is a solution of the point equation of the net N x . Similarly, any Levy 
transform N ff of the net N x along the v-tangents of N x is generated by the point 
a given by 

(82) (T = 6 v x-dx v , 

where is a solution of the point equation of N x . 

If in equations (81), (82) we take the same function 8 and then place 



we can easily show that these expressions for h, k satisfy equations (77). 
Therefore two Levy transforms of a net along the two congruences of tangents of 
the net by means of the same solution of the point equation of the net are the 
focal nets of a congruence harmonic to the net. Conversely, any congruence 
harmonic to a net can be so regarded. 

It is easy to demonstrate that if a net is conjugate to a congruence, one 
focal net of the congruence is harmonic to one of the congruences of tangents of 
the first net and the other focal net is harmonic to the other congruence of tan- 
gents. For this purpose let us consider a congruence IV represented by 
equations (69), and a conjugate net N x defined by equation (71) subject to 
the conditions (72). The first of equations (69) says that the w-tangent at 
the point Pr, passes through the point Pf, as shown in Figure 24, and hence 
through the point P x . Since we have 

(83) xi = x v -[d+ (log X) v ]x = M fo, - (log X),ij] , 

it follows that the v-tangent at Pr, passes through the point xi. The demon- 
stration that the congruence of v-tangents of the net N x is harmonic to N n 
is completed by observing that x u is a linear combination of x\ and x. 
Similarly, it can be demonstrated that the congruence of w-tangents of the 
net N x is harmonic to the net Nf. 

Two nets that are conjugate to one congruence are harmonic to another con- 
gruence. For, in the first place, it is geometrically clear that the tangent 
planes of the two nets at corresponding points, that is, points on the same 
generator of the common conjugate congruence, intersect in a line, since the 
tangent lines of corresponding curves at corresponding points of the two 
nets are coplanar. Analytic expressions for the coordinates of the two points 



32] 



CONJUGATE NETS 



155 



in which corresponding tangent lines intersect can be found in the following 
way. Let us consider a congruence IV represented by equations (69), and 




FIG. 24 

a net N x defined by equation (71) subject to the conditions (72), together 
with a net N v defined by 

(84) 2/=A-X'f 

with X', // also satisfying (72), it being supposed that the determinant D 
defined by 

(85) D = (jL\ f //X 

is not zero. Simultaneous solution of equations (71) and (84) for 77, f gives 

(86) X'# \y=Dr), n'x ny = D ; 



156 PROJECTIVE DIFFERENTIAL GEOMETRY 

and from the first of equations (73) together with the same equation with 
y, X', n' in place of x, X, M respectively we find, by elimination of , the rela- 
tion 

Comparison of the two expressions for 77 in (87) and the first of equations 
(86) gives, after some reduction, the first of the following equations, the 
second being deduced similarly, or written by symmetry: 

(88) ( X'fe.- dog )*] = %- dog D) u y] , 

I M'[* - (log D)#] = n[y v - (log D),y] . 

Let us place 
(89) 



Then equations (88) become 

(90) x'u = my' u , x' v = nyi . 

The w-tangents at the points P x , P y intersect at the point x' U) and the v- 
tangents at x' v . From equations (90), on placing p = i/i, a y' v and calculat- 
ing (x'u/m) v = (xi/ri) u , we obtain the equations 

(m n)p v = (mn)(Tu= m v p-\-n u ff , 

which tell us that the line l^ generates a congruence harmonic to both of the 
nets 'N x and N y . Thus the demonstration is completed. Figure 25 illus- 
trates the situation in the foregoing discussion. 

We shall next determine all nets that are harmonic to a given congruence. 
For this purpose let us consider a congruence IV represented by equations 
(69), and let us seek to determine all nets N x such that x satisfies two equa- 
tions of the form 

(91) x u = px+r , x v = qx+srj , 

which assert that the it-tangent at the point x passes through the point 
f , and that the v-tangent at the point x passes through the point 77. Equat- 
ing the two expressions for x uv derivable from (91) leads, after replacing 



132] 



CONJUGATE NETS 



157 



x u , x v , r? w , f v by their values, to a linear equation in z, 77, f , all of whose 
coefficients must vanish. Thus one obtains for equations (91) three in- 




FIG. 25 



tegrability conditions, one of which is p v = q u . Consequently a transforma- 
tion x = X 



can be made which will reduce both of p and q to zero. Then equations 

(91) become 

(92) z w = rf, x v = srj , 

The two integrability conditions for these equations are 

(93) r v = 8r + @s , s u = yr as . 

Our conclusion may be stated as follows. Every net N x harmonic to the con- 
gruence IV represented by equations (69) is determined by equations (92) sub- 
ject to the conditions (93). We remark that to every pair of solutions r, s of 
(93) there correspond infinitely many nets N x harmonic to IV, since x is 



158 PROJECTIVE DIFFERENTIAL GEOMETRY 

determined by equations (92) except for an additive constant, when a pair 
of solutions r, s is known. When r s = the point x is fixed and does not 
generate a proper net. 

It is now easy to show that two nets that are harmonic to one congruence are 
conjugate to another congruence. To this end let us consider two nets N x , N y , 
the first being given by equations (92) subject to the conditions (93) and 
the second by the same equations with r', s' in place of r, s. Both of these 
nets are harmonic to the congruence IV represented by equation (69). We 
suppose rsr's'^O and find 

(94) yu = mx u , y v = nx v (m = r'/r , n = s'/s) , 
so that we have 

(95) (y mx) u = m u x , (ynx) v = n v x . 

Consequently the net N x is conjugate to the congruence whose focal nets 
are generated by the points y mx, y nx. Similarly, the net N v is shown to 
be conjugate to the same congruence by writing equations (94) in the form 

[(y-mx)/m] u = -ra tt 2//m 2 , [(y-nx)/n] v = -n v y/n 2 . 

The reader may discuss the special case in which r' = r, s' = s. 

It remains to determine all congruences that are conjugate to a given net. 
Let us consider a net N x with x satisfying equation (1), and let us seek to 
determine the focal nets N^ Nf of a congruence conjugate to N x . Let N v 
be the focal net that is harmonic to the congruence xx\ ; then we can write 



(96) r? = tx+fa , rj v = <pxi+ drj . 

Equating the two expressions for rj uv derivable from these equations, and 
replacing T/ U , r/ v , Xi u , x v by expressions equivalent to them, we obtain a 
linear equation in x, Xi, ?;, all of whose coefficients must vanish. Thus we 
find for equations (96) three integrability conditions, one of which is ft = 5 tt . 
Consequently a transformation 77 = AT; with 



can be made which will reduce both of ft and 5 to zero. Then equations 

(96) become 

(97) r) u = \l/x , rj v = (f>Xi. 



33] CONJUGATE NETS 159 

The two integrability conditions of these equations are 
(98) * 



Elimination of \l/ shows that the function <p is a solution of the adjoint of 
equation (1), namely, 

(99) <pw = (ca u 'b v )(pa<f> u ~bv v . 

Therefore one focal net Nr, of any congruence conjugate to the net N x repre- 
sented by equation (1) is determined by equations (97) wherein the function <p 
is a solution of equation (99) and \l/ is determined by the first of (98). The other 
focal net N{ of the congruence is given by 

(100) f = !?-*, 

as is shown from the second of the following equations which are readily 
derivable from (100) in the presence of equations (97), (98) : 



When <p \// = the points 17 and f coincide in a fixed point and a congruence 
IV is a bundle of lines. 

33. Polar sequences of Laplace. When a sequence of Laplace and a 
hyperquadric are given in a linear space of n dimensions S n , there is deter- 
mined another sequence in the space S n by means of the polar relation 
between point and hyperplane with respect to the hyperquadric. The rela- 
tion between the two sequences is a reciprocal one, and the sequences are 
called polar sequences with respect to the hyperquadric. Such sequences 
will now be studied analytically. The exposition will follow very closely 
that of Tzitz&ca, but the fundamental ideas are due to Darboux. 

Let us consider* an unlimited sequence of Laplace immersed in a space 
S n and determined by a net N x with x satisfying equation (1). Let us also 
consider in the space S n a non-singular hyperquadric Q n -i. The equation of 
Q n _i can be written in the canonical form Sx 2 = 0, the summation extending 
from 1 to n+1. The points x\, x z , . . . , x n , which generate nets of the La- 
place sequence, determine a hyperplane $ n -i; the coordinates y of the pole 
of this hyperplane with respect to the hyperquadric Q n -i satisfy the equa- 
tions 



(102) 

* Tzitzdica, 1924. 3, p. 121. 



160 PROJECTIVE DIFFERENTIAL GEOMETRY 

and, if the proportionality factor of y is suitably chosen, also satisfy 
(103) 



Similarly, the points x, Xi, . . . , z_i determine a hyperplane whose pole 
with respect to the hyperquadric Q n -i has coordinates yi satisfying 



(104) St/!o: = , 2tyi$i = 0, . . . , S2/iZ n -i = , 

and, with proper choice of proportionality factor, also satisfying 
(105) 



If we regard the space S n as determined by the points x, Xi, . . . 9 x n , we 
can write 



(106) x- l =dx+d 1 x 1 + . . . +d n x n , 

where the coefficients d, di, . . . , d n are scalar functions of u, v. 

We shall deduce some analytic consequences of the equations just 
written and use them to prove that the point y describes a net N v when u, v 
vary, and that the point y\ also describes a net which, as the notation indicates, 
is the first Laplace transform of the net N y . Thus a ^/-sequence is defined which 
is called the polar sequence of the original z-sequence with respect to the 
hyperquadric Q_i. For the purpose of the proof it is sufficient to show that 
there exist four scalar functions p, q, r, soiu,v such that 

(107) y* = py+qyi , yiu = ry+syi . 

In preparation for the existence proof we observe that equations (102), 
(103), (106) imply 



(108) 

Differentiation of equation (103) and use of equations (2), (102), (103) give 

(109) 2y v x=-a. 

Differentiation of equation (108) and substitution for x-i v lead to 

(110) 2y v X-i = d v -ad-K . 



33] CONJUGATE NETS 161 

Finally, differentiation of 2yz r = 0(r=l, . . . , n I) and substitution for 
x r v from the first of equations (22) yield 

(111) 

How the n+l equations 



in the n+l unknown values of the expression within parentheses are such 
that the determinant of their coefficients is not zero, since the vanishing of 
this determinant would imply the linear dependence of the points z_i, x, . . . , 
x n -i. Hence there exist two functions p, q of u, v, such that the first of equa- 
tions (107) is satisfied. In fact, it is easy to verify by means of the equations 
satisfied by y v , y, 2/1 that the actual formulas for p, q are 

(112) P=-, q = d v -K. 

The demonstration for the second of equations (107) can be made simi- 
larly. 

Geometrically, the net N x which was used to determine the z-sequence 
was any net whatever of the sequence. Therefore we can generalize the 
result just obtained and say that the pole y r of the hyper plane determined by 
the points z_ r +i, X- r +2, , X- r + n is the rth transform of the net N v in the 
positive direction if r > 0, and in the negative direction ifr<Q. 

The foregoing observation permits us to demonstrate that the polar rela- 
tion between the two sequences is reciprocal, by showing that the x-sequence is 
the polar of the y-sequence. If we look at equations (104), which are satisfied 
by 2/1, and write the analogous equations which are satisfied by each of 2/2, 
2/3, . . . , 2/> namely, 



we observe that among these equations appear 
(114) 



162 PROJECTIVE DIFFERENTIAL GEOMETRY 

Therefore the point x is the pole of the hyperplane determined by the points 
2/i> - j 2/n, just as the point y is the pole of the hyperplane determined by 
the points x\, . . . , x n ; and the argument previously made can be applied, 
with x and y interchanged, to establish the desired result. 

EXERCISES 
1. If #=0, then Xi= Ve^ bdu and consequently the general solution of equation 



where U and V are arbitrary functions of u alone and v alone respectively. In gen- 
eral if H r K-8 = Q, where r, s are any positive integers or zero, equation (1) can be 
integrated by quadratures. 

2. If x satisfies equation (1) and also x vv = qx+dx v , calculate the integrability 
conditions for this system of two equations, and hence show that the locus of the 
point a: is a developable surface with its f-curves for generators, these lines being 
tangents of a w-curve into which the net Ni degenerates. 

3. The x in the parametric vector equation (II, 11) of any developable surface 
satisfies two equations of the form (10), namely, x u u = 0, x u t -\-uxu x = 0. Conversely 
if a surface S whose parametric vector equation is x = x(u, v) in space S n (n>3) is 
such that x satisfies two essentially distinct equations of the form (10), then S is 
developable. Every surface in space $3 is an integral surface of two such equations. 

SEGRE, 1907. 2, p. 1054 

4. On a developable surface in space S n the generators and any one-parameter 
family of curves form a conjugate net. 

5. If two Laplace equations L(x) = Q and L(x)=Q have their corresponding in- 
variants equal, so that H = H, K = K, then there exists a transformation x = \x 
that will convert one of these equations into the other. 

6. Use Green's characterization of plane nets with equal invariants, together 
with the facts that when the Ti-curves on a surface S in space /S 3 are indeterminate 
the reciprocal congruence T2 is harmonic to S, and that the w-tangent at each point 
on the surface S intersects the corresponding line l\ as does also the corresponding 
v-tangent on the surface S p , to demonstrate synthetically the theorem of Koenigs 
on plane nets with equal invariants. GREEN, 1919. 1, p. 106 

7. Study the effect of the transformation 

* = X, u=U(u) , v=V(v) (Xtf'WO) 

on equations (26). In particular, by means of a transformation x X#, reduce these 
equations to the canonical form for which a-f-6=0, 5-f a=0. Determine the most 
general transformation preserving this canonical form. 

WILCZYNSKI, 1911. 2, p. 477 



CONJUGATE NETS 163 

8. Prove that the differential equation of the associate conjugate net of the para- 
metric conjugate net on a surface in space 83 is Ldu*Ndv z = Q, using the defini- 
tion that the associate tangents at each point P x separate the parametric tangents 
harmonically. Prove also that the differential equation of the anti-ray net of the 
parametric net is 

LHdu?-2nLNdudv-NKdv* = Q , 

using the definition that the anti-ray tangents at P x are the harmonic reflections of 
the ray tangents in the parametric tangents. The associate tangents, the anti-ray 
tangents, and the axis tangents belong to the same involution in case n(log r) U v Q. 
Then the parametric net is either harmonic, or isothermally conjugate, or both. 

GREEN, 1916. 1, p. 313 

9. (A J net is a conjugate net in space 83 which is isothermally conjugate and 
has equal point invariants.) If a surface has on it a J net, then, by choosing the 
asymptotic parameters of Chapter III so that the / net has the equation dv*du 2 = 
0, show that /3 U = TV. For a / net the ray and associate ray intersect on the second 
edge e 2 of Green (see Ex. 26 of Chap. III)'. 

FUBINI and CECH, 1926. 1, p. 106; CECH, 1929. 4, p. 1333 

10. A focal plane of the axis at a point P x of a conjugate net in space $3 is tangent 
(at the corresponding focal point) to both ruled surfaces of axes that contain the 
curves of the net through P x . A variable plane through the axis touches these two 
ruled surfaces in a pair of points in a projectivity which has the focal points of the 
axis for double points. For the parametric net associated with system (31) the in- 
variant of this projectivity is the ratio of the roots of equation (35). 

11. The axis curves (35) of a net N x are indeterminate in case J^=n=J^==0. If 
not indeterminate they coincide with the fundamental net N x in case J^= ^f 0. In 
both cases N x consists of plane curves, so that the nets Ni and N-i are on develop- 
ables; in the first case the nets N\ and N-\ are on cones with a common vertex, 
through which all of the axes of the net N x pass. Dually, the ray curves (48) are 
indeterminate in case 7/ = n=l=0. If not indeterminate they coincide with N x in 
case H = K = 0. In both cases N x consists of cone curves, so that Ni and N-\ reduce 
to curves ; in the first case these curves are plane curves in the same plane, in which 
all the rays of the net N x lie. 

12. If the rays of a net in space S n form a congruence in the sense of Section 28, 
then either H=K=Q, the number of dimensions n being unrestricted, or else n=2 
or n = 3. 

13. Eliminating y from equations (31), calculate the coefficients of the equations 
used by Green for the study of a conjugate net, which can be written in the form 

Xuu - ax vv -f bxu -f cxv + dx , 



164 PROJECTIVE DIFFERENTIAL GEOMETRY 

Hence prove that the invariants $8', (', J) defined in Green's notation by 

8<E' = 4c'-2&+(log a) , 



are given in the notation of system (31) by 

8(' = 4&-2a-(log r) u , 



GREEN, 1915. 3 and 1916. 1 

14. In conjugate parameters the differential equation of the curves of Darboux 
(see Ex. 13) is 



and the equation of the curves of Segre is 

58 W+ 3r&du*dv - Zrft'dudv* - rg'tto = . 

If the parametric conjugate net of the present chapter is the net dv 2 X 2 dtt 2 = in 
the asymptotic parameters of Chapter III, prove that the equations $$' = and (' = 
are respectively equivalent to the equations /sdw 3 4-7^ = and pdu?ydv* = Q in 
asymptotic parameters. LANE, 1922. 3, p. 287, and 1926. 4, p. 163 

15. If a surface has on it an 12 net, then, by choosing the asymptotic parameters 
of Chapter III so that the R net has the equation dv 2 du? Q, show that /9 v = 7u 
(see Ex. 22 of Chap. V). 

16. Using the fact that a conjugate net is harmonic in case the foci of each ray 
separate the ray-points thereon harmonically, prove that the net dv 2 \ 2 du 2 = Q in 
asymptotic parameters is harmonic in case F-f GX 2 = 0, the definitions of F, G being 
given in equations (III, 37) in which a, b are given by equations (III, 68). 

17. Two congruences harmonic to one net are conjugate to another net (called 
a derived net of the first). If the congruences are generated by the line p<r given by 
equations (81), (82) and the line p', </ given by the same equations with 0' in place 
of e, the derived net is generated by the point X defined by 

x x u x v 

/JO/) 

t/u Vv 

6 6 U 6 V 

18. Two congruences conjugate to one net are harmonic to another net (called a 
derivant net of the first). If the congruences are generated by the line rtf given by 



CONJUGATE NETS 165 

equations (97), (100) and the line 17', r' given by the same equations with / in place 
of <p, the derivant net is generated by the point Y defined by 



EISENHAET, 1923. 3, p. 26 
19. If is defined by equation (50) and y by 

ri = e-o(x-i, y, zO , 

then , i) satisfy a system of equations of the form (31), of which the second is equa- 
tion (51) and of which the first and third are 



where 

p' = 4[&-+(log L/A) u ]-nL ; a'= -a+ (log LN)u , 
q' = B[a-8+(\og N/B) v ]+nN , 5'= -5 



20. The projection of a conjugate net from a point onto a hyperplane is a con- 
jugate net, and the projection of a congruence from a point onto a hyperplane is a 
congruence. 

21. If a conjugate net is immersed in space S n , and if a congruence is in a hyper- 
plane S n -i and is such that there is just one of its lines in each tangent plane of the 
net, then the congruence is harmonic to the net. 

22. Carry out an investigation analogous to that indicated in Exercise 31 of 
Chapter III, Starting with the parametric conjugate net of equations (31) instead 
of with the asymptotic net. Consider, in particular, a bundle of nets on an integral 
surface S of equations (31), consisting of all nets every one of which has the property 
that at every point P x of S its tangents separate the parametric conjugate tangents 
harmonically, and consider a pencil of nets in this bundle, determined by an arbi- 
trary net of the bundle. Prove that the rays of a point P x with respect to all the nets 
of this pencil form a flat pencil of lines. Find the coordinates of the center of this 
pencil. LANE, 1926. 4, p. 164 

23. Use equation (71) with X, unrestricted and with rj, f satisfying equations 
(69) to prove that an unspecialized transversal surface of an arbitrary congruence 
in space S n is an integral surface of a pair of differential equations of the form 

Xuuv = axuu H- hxuv -f Ixu + mx v + dx , 

m'xv+d'x . 

P. TZITZEICA, 1927. 2, p. 582 



166 PROJECTIVE DIFFERENTIAL GEOMETRY 

24. Every integral surface of a completely integrable system of equations of the 
form appearing in Exercise 23 is a transversal surface of a congruence. 

B. SEGRE, 1927. 12 

25. Calculate the effect of the transformation (59) on the coefficients of system 
(31). Prove that when n^Q,it is possible to choose & so that after the transforma- 
tion one has n= 1. Show that it is always possible to reduce system (31) to a canoni- 
cal form for which 

&-t-a+A=0, a+5-H5=0 

26. In the notation of equations (67) prove that 

2rt\ 3l> , J#i = 2J& e^fc (log Cfa )uv , 

8{ = 8$'+ (log H*jf/i*) v , 86J -SE'- (log 



27. If a sequence of Laplace is periodic of period k so that the net Nk coincides 
with the net N x , and k is numerically the smallest integer for which this happens, 
then the sequence is in a linear space of fc 1 dimensions. 

28. In ordinary space the ^-curves of a conjugate net are plane curves in case 
J^=0 and the ^-curves are plane in case J^='0. 

29. Three consecutive w-tangents constructed at a point x and two consecutive 
points of a curve C x on an integral surface of system (31) determine a quadric Q u 
whose equation referred to the tetrahedron x, x-i, Xi, y is 



) =0 
Write the equation of the quadric Q v and study the relations of these quadrics. 



CHAPTER V 
TRANSFORMATIONS OF SURFACES 

Introduction. A transformation between two geometrical configurations 
is a one-to-one correspondence between their generating elements. From 
this point of view a transformation between two surfaces regarded as point 
loci is a one-to-one correspondence between their generating points. We 
may speak of a point transformation between two nets of curves on two 
surfaces, meaning by this locution a point transformation between the sur- 
faces sustaining the nets. Some transformations of this kind have already 
occurred in this book. One instance in Section 28 is the transformation of 
Laplace between the focal nets of a congruence, two corresponding points 
being the focal points of a generator of the congruence. Another instance in 
Section 32 is the transformation of Levy between a focal net of a congru- 
ence and any net conjugate to the congruence. 

In this chapter several other transformations of surfaces will be discussed. 
The first of these is the fundamental transformation, or transformation F t 
of Jonas and Eisenhart, which is the transformation between two conjugate 
nets that are conjugate to the same congruence, two corresponding points 
being on the same generator of the congruence. The transformation of 
Koenigs to be considered in Section 35, and the transformation of Ribaucour 
in Section 39, are special cases of this transformation. In Section 36 the 
general analvtic point transformation between two analytic surfaces is in- 
troduced, and the configuration composed of two surfaces in the relation 
of such a transformation and having the additional property that their 
tangent planes at corresponding points intersect in straight lines is briefly 
studied. These considerations are further specialized in Section 37 by re- 
stricting both surfaces to be in the same ordinary space 83. In Section 38 
the emphasis is on the correspondence itself rather than on the configura- 
tion composed of the two surfaces related by the correspondence, and in 
this section the two surfaces are supposed to be in two ordinary spaces 3 , 
not necessarily the same, one of the surfaces lying in one of the spaces and 
one in the other. 

It is well known that ordinary ruled space, that is, ordinary space re- 
garded as made up of its straight lines, is four-dimensional. Section 40 is 
devoted to an account of a correspondence between the lines of this space 
and the points of a hyperquadric in space S$. In the next section this 
transformation is applied to the theory of surfaces in ordinary ruled space. 

167 



168 PROJECTIVE DIFFERENTIAL GEOMETRY 

In Section 42 Fubini's theory of W congruences is summarized to prepare 
the way for the study in the final section of Terracini's theory of W con- 
gruences in ordinary ruled space. 

34. The fundamental transformation. We begin with some definitions. 
The fundamental transformation, or transformation F, between two surfaces in 
space S n is a one-to-one correspondence between their points such that the lines 
joining corresponding points form a congruence whose developabks intersect 
both surfaces in conjugate nets which are conjugate to the congruence. The sur- 
faces cannot be focal surfaces of the congruence. It is possible also to say 
that the correspondence thus established between two conjugate nets by 
means of a common conjugate congruence is a transformation F between 
the nets themselves as well as between the surfaces sustaining them. The 
nets are called F transforms of each other, and are sometimes said to be in 
relation F. It was shown in Section 32 that if two conjugate nets are con- 
jugate to one congruence they are harmonic to another congruence, and 
conversely; the first congruence will be called the conjugate congruence, and 
the second the harmonic congruence, of the transformation F involved. If 
the conjugate congruence is a bundle of lines the two nets are said to be 
radially related, and if the harmonic congruence is in a hyperplane the two 
nets are said to be perspective. 

Several special cases of the transformation F have been considered frqm 
time to time by various geometers. The general transformation seems to 
have been studied* first by Jonas and Eisenhart. The reader who wishes 
to see a more complete treatment of this transformation than that offered 
here should consult the bookf by Eisenhart bearing the same title as this 
chapter. For instance, Eisenhart devotes considerable attention to the so- 
called theorems of permutability which are scarcely more than mentioned 
here. The point of view adopted by Eisenhart is predominantly that of 
metric geometry. 

In this section all nets that are F transforms of a given net are deter- 
mined analytically by very simple formulas. Some of the geometrical rela- 
tions of a net and an F transform of it are deduced. Two cross ratios called 
the conjugate invariant and the harmonic invariant of the transformation are 
defined. The section closes with a glance at the completely integrable sys- 
tem of partial differential equations that define, except for a projective 
transformation, a pair of surfaces in relation F in ordinary space. 

The first problem is to determine analytically all nets that are F transforms 
of a given net. For this purpose let us consider in space S n a net N x gener- 

* Jonas, 1915. 4; Eisenhart, 1917. 3; 1923. 3, pp. 34 f. 
t Eisenhart, 1923. 3. 



34] TRANSFORMATIONS OF SURFACES 169 

ated by a point P z whose coordinates x are functions of two independent 
variables u, v and satisfy an equation of Laplace, 

(1) x uv = cx+ax u +bx v . 

Equations (IV, 81), (IV, 82) show that any congruence pa harmonic to the 
net NX consists of lines crossing the tangents of N x in the points p, a de- 
fined by 

(2) P=(*/0), er=(x/0),, 

where 6 is a solution of equation (1). The same result can also be reached by 
observing that equation (1) can be written in the form 



(3) 

wherein A, B are defined by 

(4) A=a-(log0), , = &-(log0) M . 

If a second net N v is also harmonic to the congruence po-, the proportionality 
factor of y can be chosen so that each of the lines pa crosses the correspond- 
ing tangents of N y in the points y u , y v . Since the points p, y u coincide at one 
focal point of the line pa-, and the points a, y v at the other, it follows that 

(5) 2/ w = w(z/0)u , y v = n(x/d) v , 

where m, n are scalar functions of u, v which are restricted to satisfy the 
integrability conditions that result from equating the two expressions for 
y uv derivable from equations (5) in the presence of equation (3), namely, 

(6) m v = A(n m) , n u = B(mn) . 

The point equation of the net N v is (y u /m) v = (y v /ri) u , and can be reduced to 

(7) y u * = Any u /m+Bmy v /n . 

The solution of the problem at hand can now be formulated in the following 
theorem: 

When a net N x with the point equation (I) is given, any net N v which is an F 
transform of N x can be determined by first finding a solution 6 of equation (1), 
by next calculating A, B in (4), by then solving equations (6) for m, n, and by 
finally performing the quadrature indicated by equations (5) for y. 



170 



PROJECTIVE DIFFERENTIAL GEOMETRY 



A more general analytic representation of a transformation F than the 
one just explained results if no restriction on the proportionality factor of y 
is made. In order to write the equations analogous in this case to equations 




FIG. 26 



(1), . . . , (7), it is sufficient to replace y wherever it appears in the equations of 
the preceding paragraph by y/<p, where p is a solution of the point equation of 
the net N y . On the other hand, a simpler analytic representation than 
either of these can be obtained by supposing that the proportionality factor 



34] TRANSFORMATIONS OF SURFACES 171 

of x has been chosen so that the harmonic congruence of the transformation 
consists of lines crossing the tangents of the net N x in the points x u , x v . 
This simplification amounts to taking 6=1, and hence c=0, in the preced- 
ing paragraph. The formulas employed in this analytic representation of a 
transformation F, which is in a sense the simplest analytic representation 
possible, are written together below for convenience in the order in which 
they are used in passing from the net N x to the net N y : 

x u v ax u ~\~bx v , 

(8) - ^^mx'^ 'y*^ b(m ~ n] ' 

y uv = any u /m-{-bmy v /n . 

Equations (8) will now be employed to deduce some geometrical relations 
among the various parts of the configuration composed of two nets in rela- 
tion F together with the associated conjugate and harmonic congruences 
(see Fig. 26). That the focal points of a line l xy are the points P^ PS defined 
by 

is readily verified after observing that differentiation gives 

(10) 

1 " ' - 1 - - '--n v x , 

the Laplace transformed points xi, X-\ of the point P x being defined by the 
usual formulas, 

(in rri=r fir r i=r by 

\^L L J */! t'v U/JU , */ 1 JL/u \JJj . 

The Laplace transformed points y, y-\ of the point P y are given by 

The point equations of the nets JV,, Nf are respectively 
(13) 



172 PROJECTIVE DIFFERENTIAL GEOMETRY 

where H, K are the Laplace-Darboux invariants of the net N x which were 
defined by formulas (IV, 7). Hence the Laplace transforms TJI, i/_i, fi, f_i 
of the points P,, Pj- are given by 

fra a i7i=(tt m)(Hrj+m u Xi) , (n 
I (w-n)fi = n v T7 , n v _i = (ra 

Therefore the points x\, yi, 17, rji are collinear, as are also the points X-i, t/_i, 
f , f_i. The first focal plane of the line l xv , which is tangent to the net TV, 
at the point P,, contains the points x\, 2/1, 771 in addition to the points x, y y 17, 
; the second focal plane contains the points #_i, y-i, f_i. 

There are two cross ratios of particular interest associated with a funda- 
mental transformation. The first of these is the cross ratio of two corre- 
sponding points x, y and the two focal points 77, f on the line xy. This is 
denoted by R and is called the conjugate invariant of the transformation. 
By means of equations (9) we find 

(15) R = n/m . 

As an application let us consider the special case R 1. In this case m = n 
and equations (9) show that the focal points 77, coincide. Moreover, the 
second and third of equations (8) show that m = n = const. Equations (10) 
show that the point 77 is fixed. Therefore the conjugate congruence xy of 
the transformation is a bundle of lines, and the nets N x , N y are radially 
related; the point equations of these nets, namely, the first and last of (8) 
are identical. The converse reasoning may be supplied by the reader. The 
special case R = 1 will be discussed in the next section. 

The second cross ratio referred to above is that of the tangent planes of 
the nets N x , N v at two corresponding points x t y and the first and second 
focal planes of the corresponding generator of the harmonic congruence, the 
first focal plane being tangent at the point x v to the locus of this point and 
the second focal plane bearing the same relation to the point x u . This cross 
ratio is denoted by S and is called the harmonic invariant of the trans- 
formation. 

// the configuration composed of two nets N x , N v in relation F is in ordinary 
space 83, then the four coordinates x and the four coordinates y form four pairs 
of solutions of a completely integrable system of differential equations which can 
be reduced* to the form 



x uv = ax u +bx v , 
x vv =qx+y 
2/ tt =mx u , 
* Lane, 1929. 1, p. 460. 



35] TRANSFORMATIONS OF SURFACES 173 

The reduction does not differ essentially from that employed in obtaining 
equations (8), except that one observes that the space S* may be regarded 
as determined by the points z, x U) x v , y, and that the points x vu , x vv are cer- 
tainly linearly dependent on these points. The coefficients of equations (16) 
satisfy the following integrability conditions, which are found by the usual 
method from the equations 

(x uu )v = (x uv ) u , (x uv ) v = (x vv ) u , (y u )v = (y v )u 
and the fact that the points x t x u , x v , y are not coplanar: 



(17) 



v a(nm) , n u = b(mri) . 

Not only the original pair of nets but also any pair into which this pair can 
be projected gives rise to the same system (16). Conversely, equations (16) 
with the integrability conditions (17) satisfied define a pair of nets in relation 
F in space 83, except for a projective transformation. 

The conjugate invariant R of a pair of nets as just defined in space SB is 
still given by (15) ; the harmonic invariant S can be calculated by observing 
that the first and second focal planes of a line x v x u of the harmonic con- 
gruence intersect the corresponding line xy in the points qx+Ny, px-\-Ly 
respectively. The result of the calculation is 

(18) S = pN/qL . 

Additional results on fundamental transformations in ordinary space may 
be found in Exercises 3, 4, 25. 

35. The conies and transformation of Koenigs. Conjugate nets with 
equal Laplace-Darboux invariants Hj K constitute a class of nets of con- 
siderable interest. Plane nets with equal invariants were characterized 
geometrically in Section 29, and conjugate nets with equal invariants in 
ordinary space were characterized in Section 31. We shall give in the pres- 
ent section a geometric characterization of conjugate nets with equal in- 
variants in space S n , using some conies called the conies of Koenigs. More- 
over, in this section we shall also consider the special kind of fundamental 
transformation called the transformation of Koenigs of one conjugate net 



174 PROJECTIVE DIFFERENTIAL GEOMETRY 

with equal invariants into another net of the same kind, and shall study 
briefly certain pencils of conies containing the conies of Koenigs at corre- 
sponding points of the two nets. 

In order to define the conies of Koenigs we make the following observa- 
tions. In the tangent plane at a point P x of a net N x in space S n , with x 
satisfying equation (1), there is a pencil of conies tangent to the curve C u 
at the point xi and tangent to the curve C v at the point X-i. In order to 




FIG. 27 

write the equation of a general conic of this pencil, let us choose the points 
x, x~i, x\ as the vertices of a local triangle of reference in the tangent plane, 
with a unit point chosen so that a point yix+yzX-i+y&i may have local 
coordinates proportional to yi, y 2 , y$. Then the equation of a general conic 
belonging to the pencil of conies tangent to the line 7/2 = at the point 
(0, 0, 1), and tangent to the line 7/3 = at the point (0, 1, 0) is of the form 
(see Fig. 27) 

(19) y\ -fy z yz = (/ arbitrary) . 

It will be shown in the next paragraph that there is one conic of this pencil 
which has three-point contact with the curve C u at the point xi, and simi- 
larly one conic of the pencil which has three-point contact with the curve 
C v at the point z_i. These two conies are by definition the conies of Koenigs 
at the point P x of the net N x . 

The equations of the conies of Koenigs can be derived in the following 
way. The coordinates of any point X near the point x\ on the curve C u 
that passes through the point xi can be represented by Taylor's expansion 



35] TRANSFORMATIONS OF SURFACES 175 

as power series in the increment Aw corresponding to displacement from the 
point xi to the point X along C u : 



Making use of equation (IV, 18) and the equation obtained therefrom by 
differentiating once with respect to u, we find 



where, to terms of as high degree as will be needed, 
(20) 2/i 



Demanding that equation (19) be satisfied by the power series (20) for 
2/i, 2/2, 2/3 identically in Aw as far as the terms of the second degree, we find 
f=2H. If in place of C u and x\ we hatf used C v and x~\ we should have 
found f=2K. Thus we obtain the equations of the conies of Koenigs, 

(21) 2/i ~ 2#2/22/3 - , 2/1 - 2^2/22/3 = . 

These conies coincide in case H = K. Thus the following theorem is estab- 
lished. 

Conjugate nets with equal invariants in space S n are characterized by the 
property that in the tangent plane at each point x of any one of them there exists 
a conic having three-point contact with the u-curve at the ray-point Xi and also 
having three-point contact with the v-curve at the ray -point x-\. 

The definition of the transformation of Koenigs can be formulated in the 
following words. Two conjugate nets N x , N v in relation F are said to corre- 
spond by a transformation of Koenigs in case the foci Pr,, PS of each line xy 
separate harmonically the points P x , P y thereon. Then R 1, and equation 
(15) shows that n m; consequently the second and third of equations 
(8) give 

(22) m v 2am , m u 2bm . 

It follows that a u = b v , and the net N x has equal invariants. Moreover, in 
this case the first and last of equations (8) show that each coefficient of the 
point equation of the net N v differs only in sign from the corresponding 
coefficient of the point equation of the net N x ; hence N v also has equal in- 
variants. So we reach the conclusion: 

Each of two conjugate nets that correspond by a transformation of Koenigs 
has equal invariants. 



176 PROJECTIVE DIFFERENTIAL GEOMETRY 

We proceed to consider certain pencils of conies associated with the trans- 
formation of Koenigs. The tangent planes at corresponding points of two 
conjugate nets N x , N v in relation F in space S n determine a space $3 in 
which the points #, x u , x v , y can be used as the vertices of a local tetrahedron 
of reference with a unit point chosen so that a point Zix+zzx u +z$x v +ziy 
may have local coordinates proportional to 21, . . . , z 4 . In this coordinate 
system the conic (19) has the equations 

(23) 2 4 = (2i + &22 + a2 3 ) 2 -/2 2 23 = , 

since the following relations hold between the coordinates of a point in the 
tangent plane of the net N x in the two systems: 

2i = 2/i-&2/2 = a2/3 , 22 = 2/2, 23 = 2/3, 2 4 = 0. 

If a point in the tangent plane of the net N y has coordinates y[, y' z , y' 6 re- 
ferred similarly to the triangle y t y~i, 2/i> the following relations exist be- 
tween these coordinates and the coordinates 21, . . . , 24 of the same point: 

21 = , 2 2 = myi , 2 3 = m/3 , 2 4 = 2/1 - bmyi/n - any' z /m . 

Therefore a general conic of the pencil in the tangent plane of the net N v , 
tangent to the curve C u at the point y\ and to C v at the point t/-i, has the 
equations 

(24) Zi=(mnzt+bmz2+anz 3 ) 2 0ran2 2 2 3 = (0 arbitrary) . 

These two pencils of conies determine involutions on the line z\ = 24 = 0, the 
double points of the first involution being obtained by solving 



and those of the second by solving 



These two involutions are the same in case ra 2 = n 2 , that is, in case the 
transformation F is either a radial transformation or else a transformation 
of Koenigs. In the latter case two conies that intersect the line 2i = 2 4 =0 
in the same pair of points have their parameters connected by the relation 

(25) 



36] TRANSFORMATIONS OF SURFACES 177 

Two such conies determine a pencil of quadric surfaces, 

(26) | 

the parameter of the pencil being A. If /(/ 4a6) 5^0, so that the two conies 
are proper conies, there are two cones besides the planes ZiZ 4 =0 in the pen- 
cil, and for them A = m (see Ex. 9) . 

36. Pairs of surfaces with their points in one-to-one correspondence. 

The configuration composed of two surfaces with their points in one-to-one 
correspondence is of frequent occurrence in differential geometry. Instances 
of this are two surfaces sustaining conjugate nets in the relation of a Laplace 
transformation, or a Levy transformation, or a transformation F. It is the 
purpose of this section to consider, in the first place, the general analytic 
point transformation between two analytic surfaces. Afterward the restriction 
is imposed that the tangent planes at corresponding points of the two 
surfaces intersect in straight lines. This restriction may, from one point of 
view, be considered as mild, since all pairs of surfaces with their points 
in correspondence in ordinary space are of necessity subjected to it, but 
from another point of view it is quite severe, since two planes in space 
S n (n>4) ordinarily do not intersect at all. 

We wish to write the equations of the general analytic point transformation 
between two analytic surfaces. In space *S n let us consider two proper analytic 
surfaces S x , S y with the respective parametric vector equations 



On the surface S x let us consider a region in which points P x are in one-to- 
one correspondence with pairs of values of u, v, and similarly for S v . Then a 
one-to-one correspondence between points P x , P y is equivalent to a func- 
tional relation between the parameters u, v and , 17 which may be expressed 
by equations of the form 

(27) = {(u, t>) , 77 = q(u, v) (fa v - Srt* * 0) , 

the functions , 17 being supposed analytic. These are the desired equations 
of transformation. 

A curve on either one of the surfaces S x , S v will be said to correspond to a 
curve on the other in case the two curves are generated by corresponding 
points. All the curves through a point P x on the surface S x have at P x 
tangent lines which form a pencil in the tangent plane of S x and with its 



178 PROJECTIVE DIFFERENTIAL GEOMETRY 

center at P x ; these curves correspond to curves through a point P v on the 
surface S v whose tangents at P v form a pencil in the tangent plane of S v 
and with its center at P y . So there is established a line correspondence 
between the pencils of tangents at corresponding points P z , P VJ corre- 
sponding lines being tangents of corresponding curves. This line correspond- 
ence will now be shown to be a projectivity. The direction dv/du of a curve 
at the point P x and the direction drj/d% of the corresponding curve at the 
point P v are connected by the linear fractional relation 

(28) drj/dt = (nu+tydv/duj/fa+tidv/du) , 

which follows immediately from equations (27). Therefore four curves at 
the point P x have tangents whose cross ratio is equal to that of the tan- 
gents of the corresponding curves at the point P v . Since the correspondence 
between the pencils of tangents is not only one-to-one but also preserves 
cross ratio, it is a projectivity, as was to be shown. 

It is possible to obtain for the point correspondence under consideration 
a simpler analytic representation than that appearing in equations (27). If 
the expressions for , t] as functions of u, v given in (27) are substituted in the 
parametric vector equation of the surface S y , then corresponding points P x , 
P v will have the same curvilinear coordinates u, v. Therefore it is no restric- 
tion on the correspondence or on the surfaces to suppose that the parameters have 
been chosen so that corresponding points have the same curvilinear coordinates. 
We shall suppose from now on that this choice of parameters has been made ; 
then, as equation (28) shows, corresponding curves will have the same direc- 
tion dv/du at corresponding points. 

A restriction will now be imposed on the correspondence itself. Let us 
suppose that the tangent planes at corresponding points P x , P v of the two sur- 
faces S x , S v intersect in straight lines. Let us further suppose that the line h of 
intersection of the tangent planes at each pair of points P x , P v passes through 
neither P x nor P y . In this case x, y satisfy two equations which can be 
written in the form 

( y u =fx +mx u +sx v +Ay , 
(^jy ) \ 

{y v = gx+tx<u +nx v +By (mnst^fy . 

For, the first of these equations asserts that the point y u Ay on the u- 
tangent at the point P v is in the tangent plane at the point P x of the surface 
S XJ and the second equation makes a similar statement involving the v- 
tangent at P v . The existence of the inequality mn s^O can be demon- 
strated by supposing that mnst = Q, eliminating x u and x v from equations 
(29), and thus proving that the line h passes through the point P X} contrary 



TRANSFORMATIONS OF SURFACES 



179 



to hypothesis. In the presence of this inequality, equations (29) can be 
solved for x u , x v as linear combinations of y, y u , y v , x and so it becomes evi- 
dent that the surfaces S x , S v are actually being treated in an essentially 
symmetrical way, although, to be sure, x and y do not enter equations (29) 
symmetrically. 




FIG. 28 



The two pencils of tangents at two corresponding points P x , P v have 
been shown to be projectively related. Hence these pencils determine on the 
line h a projectivity, which we proceed to discuss. Referring to Figure 28, 
let us consider on the surface S x a family of curves dv\du=Q, and let us 
denote by (7 X the curve of this family that passes through P z . The tangent 
of C x at P x is determined by x and x u +x v ^. There is a corresponding curve C x 
through the point P v on the surface S v . The tangent of this curve at P v 



180 PROJECTIVE DIFFERENTIAL GEOMETRY 

is determined by y and y u -}-y v \. But if the second of equations (29) is multi- 
plied by X and added to the first the result can be written in the form 

y+y.A- (A +B\)y = (f+g*)x+(m+t\)x u +(s+n\)x v . 

Here we have two equal expressions for the point in which the tangent of 
C x at P v meets the line h. The direction /x of the line joining the point P x to 
this point is the ratio of the coefficient of x v to the coefficient of x u in the 
right member of the equality above, and is therefore connected with the 
direction X by the equation 

(30) (m+X)Ai = a+nX . 

This equation represents the aforesaid projectivity on the line h, since this pro- 
jectivity is a section of the projectivity between the pencil of tangents of 
curves through the point P x and the superposed pencil of lines joining P x 
to the points where the corresponding tangents at the point P v meet the 
line h. 

The double points of the projectivity on the line h are found by setting ^ = X 
and solving the equation 

(31) a 2 +(w-n)\-s = . 

We shall confine our attention from now on to the case in which (m w) 2 + 
4s^0, that is, the case in which the double points are distinct. 

Each double point of the projectivity on the line h is the point of inter- 
section of a tangent at the point P x and the corresponding tangent at the 
point P y . This fact indicates that there are on the surface S x two one-parame- 
ter families of curves such that at every point P x the tangent of each of the two 
curves of these families intersects the tangent of the corresponding curve drawn 
at the corresponding point P v . Without any additional restriction on the 
correspondence between the surfaces S xt &v these curves can be made to 
be the parametric curves by a suitably chosen transformation of parameters 

U = <p(u, V) , V=\//(U, V) (J^putv Vvtu^O) . 

The effect of this transformation on system (29) is to produce another sys- 
tem of the same form, of which the coefficients s, I are found to be given by 



36] TRANSFORMATIONS OF SURFACES 181 

Let us choose for <p, $ two functionally independent solutions of the partial 
differential equation 



Then we shall have s = I = 0. This transformation amounts to setting s = = 
in equations (29), so that the it-tangents and y-tangents at corresponding 
points P x , P v may intersect. 

With the analytic simplification just accomplished, the invariant* of the 
projectivity on the line h, that is, the cross ratio of the invariant points and 
any two corresponding points, is found, by calculating the cross ratio 
(oo, 0, X, n), to be n/m. 

The lines l xv joining pairs of corresponding points P x , P v certainly consti- 
tute a two-parameter family; this family can be shown to be a congruence, 
in the following familiar way. One starts with an arbitrary point P e de- 
fined by 

(k scalar) 



on a line l xv ; one considers an arbitrary curve through the point P x on the 
surface S x ; one demands that the ruled surface of lines l xy intersecting S x in 
this curve be developable, and that P z be the corresponding focal point of l xy ; 
thus one readily obtains by means of equations (29) the following equations 
for determining the developables and focal surfaces of the congruence xy: 

sdu 2 (m n) dudv tdv 2 = , 
k 2 -\-(m-\-n)k -\-mn- st = . 

When s = = 0, ra^n, the developables of the congruence xy are determinate 
and intersect the surfaces S x , S v in the parametric curves thereon. The foci 
of a line l xv are then the points ?/, f given by 



We may therefore state the following theorem. 

// two surfaces S x , S v have their points in a one-to-one correspondence such 
that the tangent planes of the surfaces at each pair of corresponding points P x , P v 
intersect in a straight line (variable with P x , P v ) which passes through neither 
P x nor P y and whose projectivity has two distinct double points, then the lines l xy 
joining pairs of corresponding points form a congruence. 

The converse theorem that two transversal surfaces of a congruence are 
such that the tangent planes of the two surfaces, at the two points where 

* Graustein, 1926. 5. 



182 PROJECTIVE DIFFERENTIAL GEOMETRY 

any generator of the congruence pierces them, intersect in a straight line is 
readily demonstrated geometrically. It is sufficient to observe that each 
of the two developables of the congruence containing the generator inter- 
sects the surfaces in curves whose tangents at the points on the generator 
are coplanar. 

The condition that the tangent planes at corresponding points of two 
surfaces S x , S y intersect in lines on each of which the projectivity has two 
distinct double points has just been shown to be equivalent to the condition 
that S x , S v be transversal surfaces of a congruence, and hence specializes, or 
imposes a restriction on, these surfaces. In fact, if the parametric curves on S x 
do not form a conjugate net when s = t = Q,it can be shown that S x is an integral 
surface of a system of two partial differential equations of the third order (see 
Ex. 23 of Chap. IV), and similarly for S y . To find these equations for the 
surface S x let us take s = Z = in equations (29) and differentiate the first 
of these equations partially with respect to v and the second with respect 
to u and subtract. If A v = B u , the result is a Laplace equation for x, so that 
the parametric curves on the surface S x form a conjugate net. If A v B u = 0, 
the result is 

(32) y = Px+Qx u +Rx v +Sx uv , 

where the coefficients P, Q, R, S are defined by 

3 U A v )P=f v Qu+gA fB , (B u A v )Q = m v mB g , 



(33) 

1 /T ^ '"" (B u -A v )R=-n u +nA+f. 



Differentiating equation (32) and eliminating y u , y v , y by means of equa- 
tions (29), (32) we obtain the aforementioned equations of the third order, 

uuv = (AP+f-P u )x+(AQ+m-P-Q u )x u +(AR-Ru)x v 

+ (AS-S u -R)x uv -Qx uu , 
(34) ^ 

+ (BS-S v -Q)x uv -Rx vv . 



The integrability conditions of system (34), which are obtained from the 
equation (x uuv ) v = (x uvv ) Uf can be shown by means of equations (33) to be 
satisfied identically. It is also worthy of remark that the space S(3, 0) 



37] TRANSFORMATIONS OF SURFACES 183 

at each point P x of the surface S x (defined to be the ambient of the osculat- 
ing space S 3 at P x of every curve on S x through P x ) is at most a space S 7 
instead of being the usual space Sg, since equations (34) are two relations 
among the ten points 

X, Xuj Xyj X-uuj %uv) Xvv) %uuu) %uuv) Euvvj X^vv 

which determine the space 5(3, 0). 

37. Pairs of surfaces in 3. Since in ordinary space $ 3 two distinct planes 
always intersect in a straight line, it follows that the general analytic 
point transformation between two analytic surfaces in space *S 3 is of the 
type considered in the latter part of the preceding section. It is the purpose 
of the present section to lay the foundations for the study of the configuration 
composed of two surfaces in the same space 83 with their points in one-to-one 
correspondence. The completely integrable system of partial differential 
equations employed for this purpose will be of a relatively general nature. 
The reader should presently observe for himself that several of the systems 
of equations that are fundamental for theories that have been developed 
earlier in this book are special cases of the system used here. The notation 
will be chosen with a view of exhibiting the relations of these systems of 
equations. 

We now establish the basic system of differential equations for the configu- 
ration under consideration, and write the integrability conditions therefor. 
If two surfaces S x , S y in ordinary space have their points in a one-to-one 
correspondence, such that corresponding points P x , P v have the same 
curvilinear coordinates w, v, and such that each point P v does not lie in the 
tangent plane of S x at the corresponding point P X) then S x , S v are a pair of 
integral surfaces of a system of differential equations of the form 



(35) 



x uv = cx+ax u +bx v +My 
x vv = qx+yx u +dx v +Ny , 
y u =fx+mx u +sx v +Ay , 
y* = gx+tx u +nx v +By 



In fact, if each pair of coordinates x, y is substituted in turn in the first of 
equations (35) with the coefficients p, a, 0, L regarded as unknown, the re- 
sult is four linear algebraic equations which can be solved uniquely for these 
coefficients. Similarly the other coefficients can be determined. The inte- 



184 



PROJECTIVE DIFFERENTIAL GEOMETRY 



grdbility conditions for this system of equations are found by the usual 
method; we have 



but there can exist no linear relation with non-vanishing coefficients con- 
necting x, Xu, x v , y, since the point y does not lie in the tangent plane at the 
point x. The resulting twelve integrability conditions can be written as 
follows: 



(36) 



a v +a?+by+tM = - 



'q u +c6+py+fN , 



= N u +AN+8M+yL . 
t u +ta+an+mB+g = i 



B u +tL+nM = A v +sN+mM . 



It is desirable to treat the surfaces S x , S y in an essentially symmetrical 
way, although x and y do not enter equations (35) symmetrically. Let us 
define a function A by placing 



and let us suppose that A?^0. This amounts to supposing that each point P x 
is not in the tangent plane of the surface S y at the corresponding point P v . 
Then x, y satisfy a system of equations of the form (35) but with the roles 
of x and y interchanged. In order to calculate the coefficients of this system 
we observe that two of the desired equations result at once from solving the 
last two of equations (35) for x u and x v . The other three of the new equations 
come from differentiating the last two of (35) and eliminating the deriva- 
tives of x. The twenty coefficients of the five equations thus calculated are 
indicated by accents and given by the following formulas: 



37] TRANSFORMATIONS OF SURFACES 

Af' = sBnA , Aw' = n , As' = s , 



185 



(37) 



nf , 



'c 22 



a = 



where the coefficients c t / are defined by placing 



Cn=n v +g+n5+bt , 



Precisely as in the preceding section the parameters can be chosen so that 
s = 2 = 0; then equations (36), (37) are materially simplified, but it will be 
left to the reader to observe just what the simplifications are. The develop- 
ables and focal surfaces of the congruence xy are determined by the same 
equations as in the preceding section. 

When s== = 0, the line h of intersection of the tangent planes at two 
corresponding points P x , P v of the surfaces S x , S y joins the points P p , P 
defined by 



(38) 



p = x u +fx/m , 



r = x v + gx/n 



as is seen on inspecting the last two of equations (35). When u, v vary, the 
line h generates a congruence p<r, whose developables and focal surfaces will 
now be determined. If, as the point P x describes a curve of the family 
dv \du = on the surface S xt the line h generates a developable of the con- 
gruence per, and if the point Pf defined by 



p -f k<r 



(k scalar) 



is the corresponding focal point of the line h, then h is tangent to the locus 
of the point PJ-; consequently the derivative f ' may be expressed as a linear 



186 PROJECTIVE DIFFERENTIAL GEOMETRY 

combination of p, a- only. But by actual calculation it is found that f ' 
appears as a linear combination of #, p, <7, ?/. Setting equal to zero the co- 
efficients of x, y therein, we obtain conditions on the functions k, X neces- 
sary and sufficient that the line h may generate a developable of the con- 
gruence pa and have Pf for focal point, namely, 

rn 
(39) \ 



Elimination of k and substitution of dv/du for X give the differential equation 
of the developables* of the congruence per, namely, 

, . f (mLM'-nML')du 2 +[m(LN'+MM')-n(NL'+MM')]dudv 
( } 



Moreover, elimination of X gives the equation for the determination of the foci 
of the line h y namely, 



(41) { 



m(MN'-NM')k*+[m(LN'-MM')-n(NL'-MM')]k 

-n(ML'-LM') = Q. 



For further discussion of this subject the reader is referred to Exercises 12, 
13, 14, 26. 

38. Point correspondence between two surfaces in different spaces 83. 

In the preceding section the two surfaces S x , S v considered were supposed 
to be in the same ordinary space 83, and the emphasis was on the configura- 
tion determined by the two surfaces and by the correspondence between 
them. In the present section one of two surfaces with their points in one- 
to-one correspondence will be supposed to be in a space $ 3 , and the other 
also in a space 83, ordinarily but not necessarily distinct from the first space. 
Moreover, the emphasis will be on the correspondence itself. 

The exposition will follow in some of its main features a memoir f of 
Bompiani. Other geometers, notably Cech and Miss Sperry, have also 
contributed to the results to be set forth, which are of the same general 
nature as the result reached near the beginning of Section 36 concerning 
the projectivity between the pencils of tangents at corresponding points 
of the two surfaces. As that result concerns the pencils of tangent lines of 
corresponding curves at corresponding points, so the results here concern 
the bundles of osculating planes of corresponding curves at corresponding 

* Grove, 1928. 2. f Bompiani, 1923. 1. 



38] TRANSFORMATIONS OF SURFACES 187 

points of the two surfaces. Near the close of this section we arrive at the 
particular point transformation between two surfaces that is called pro- 
jective applicability and was first defined by Fubini. 

We shall now establish the fundamental differential equations. Let us 
consider in a space 3 a surface S x , and in a space 83, not necessarily the 
same as the first space, a surface S y . Let the points P x , P v of these surfaces 
be in one-to-one correspondence, and let corresponding points have the 
same curvilinear coordinates u, v. Let us suppose that the parametric curves 
on the surface S x do not form a conjugate net, so that the four coordinates x 
do not satisfy an equation of Laplace. Then these coordinates are solutions 
of a completely integrable* system of differential equations of the form 



For, if the four coordinates x are substituted one at a time in the first of 
equations (42) with the coefficients p, a, ft L regarded as unknown, the 
result is four linear algebraic equations which can be solved for these co- 
efficients. The coefficients q, 7, 6, N can be determined in like manner. 
Similarly, if the parametric curves on the surface S v do not form a conjugate 
net, the four coordinates y are solutions of a system of the same form, whose 
coefficients will be indicated by accents. 

The correspondence between the points of the surfaces S x , S v determines 
a correspondence between the bundles of planes with centers at two corre- 
sponding points P x , P v . The equations of this correspondence can be 
found in the following way. Let us consider the curve C x of the family 
dv \du = through the point P x on the surface S x . The coordinates of the 
osculating plane of C x at P XJ referred to the tetrahedron x, x u , x v , x uv with 
suitably chosen unit point, are found to be given by 






The coordinates rj of the osculating plane of the corresponding curve at P v 
on S V) referred to the tetrahedron y, y u , y v , y uv with suitably chosen unit 
point, are given by similar formulas with the same X, X' but with accents on 
the coefficients. Homogeneous elimination of X, X 7 furnishes the equations of 
the correspondence between the bundles of planes with centers at corresponding 

* Green, 1920. 2. 



188 PROJECTIVE DIFFERENTIAL GEOMETRY 

points P X) Py> corresponding planes being the osculating planes of correspond- 
ing curves: 



(44) 



The equations of the inverse transformation can be obtained by keeping 
in mind that 771 =0 and solving equations (44) for the ratios of rj 2 , 173, 174, or 
more simply by merely interchanging in (44) the plane coordinates and rj 
and at the same time the accented and unaccented letters. Therefore the 
correspondence between the bundles of planes at corresponding points of the two 
surfaces is a cubic Cremona transformation. 

Considerable analytic simplification can be obtained by supposing that 
the parametric curves on the surface S v are the asymptotic curves, so that 
L'=Ar' = 0. Under this assumption let us seek for those planes at a point P x 
whose corresponding planes at the corresponding point P y are indeterminate. 
For this purpose we set ?? 2 = ^3 = ^4 = in equations (44) and solve the re- 
sulting equations for the ratios of 2, 3, 4. As the result we find three planes 
whose coordinates are 

(0, 0, 0, 1) , (0, N, 0, 7 '-7) , (0, 0, L, p-fi . 

If the asymptotic curves on the surfaces S x , S v do not correspond, then 
ZJV^O and these three planes are distinct. The first is the tangent plane at 
the point P x of the surface S x . By means of the equations of the inverse of 
the transformation (44) it can be shown that this plane corresponds to every 
plane through either of the straight lines 



which are the tangents at the point P v of the curves on the surface S v corre- 
sponding to the asymptotic curves on the surface S x . Similarly, it can 
be shown that the plane (0, N, 0, 7' 7) corresponds to every plane 
through the v-tangent, 2/4 = 2/2 = 0, at the point P v , and that the plane 
(0, 0, L, jS' j8) corresponds to every plane through the u-tangent, 2/4=2/3 = 0, 
at P v . It will be left to the reader to show that the last two planes at the 
point P x correspond to the osculating planes of curves on S v having inflex- 
ions at P y . Finally, the line of intersection of these two planes at P x passes, 
of course, through P x and passes also through the point 



[0, L(y-y'), N(ft-F), LN] . 
This line is called the axis of the correspondence at the point P x . 



I 38] 



TRANSFORMATIONS OF SURFACES 



189 



We next define a projectivity between the bundles of lines at correspond- 
ing points P x , P v of the two surfaces S x , S v . Any line l yg through the point 
P v may be regarded (see Fig. 29) as joining P v to a point P z with coordinates 





FIG. 29 



0, 22, 23, 24. To the planes of the pencil having l yz as axis correspond at the 
point P x the planes enveloping the cone whose equations, written by the 
aid of the condition of united position 2 2 i?2-f-Z3i73+24'?4 = and equations 
(44), are 



(45) 



This cone has three cusp-planes which intersect in the line l xw joining P x to 
the point P w whose coordinates 0, w 2 , w 3 , w>4 are given by 



(46) 



These equations evidently represent a projectivity between points P z 
in the plane z\ and points P w in the plane Wi 0. Projecting these planes 
from the points P x and P v respectively, we obtain two bundles of lines with 
centers at P x , P v > the equations (46) may be said to represent a projectivity 
between the lines of these bundles. 



190 PROJECTIVE DIFFERENTIAL GEOMETRY 

The projectivity (46) is degenerate if, and only if, 



Let us with Bompiani consider these two possibilities briefly and then ex- 
clude them from our future considerations. If LN = 1, the surface S* is 
developable. If LN = 4, the cross ratio of the two parametric tangents at the 
point P x (which correspond to the asymptotic tangents at the point P v ) 
and the two asymptotic tangents at P XJ in one of the possible orders, is 
( l+t'3 1/2 )/2, where t' 2 = 1. Therefore if LN = 4 the parametric tangents 
at the point P x , one asymptotic at P x , and the harmonic conjugate of the other 
asymptotic tangent with respect to the parametric tangents form an equian- 
harmonic group. 

We now define with Bompiani a certain homology in the bundle of lines at 
a point P x . Let us consider the projectivity corresponding to the pro- 
jectivity (46) when the roles of x and y are interchanged, in which a line 
xw' through the point P x , regarded as the axis of a pencil of planes, corre- 
sponds to a line yz f through the corresponding point P y , regarded as the 
cusp-axis of a cone. The equations of this projectivity are 



z' 3 = Lw' 2 -2w' 3 +(a'-a)w' 4 . 



Let us suppose that the line yz f coincides with the line yz. Then Zg, 23, 2j 
are proportional to z z , 23, 24 respectively, and these coordinates can be 
eliminated from equations (46), (47). The result of the elimination is 



(48) 



These equations evidently represent a projectivity in the bundle of lines at 
the point P x , and this projectivity can be shown to be a homology by show- 
ing that it has a flat pencil of invariant lines and one other invariant line 
through the center of the pencil. Thus the reader may demonstrate the 
following theorem. All the tangent lines at the point P x are invariant lines of 
this homology; the other invariant line is the axis of the correspondence at P x . 
The theory of union curves* can be advantageously employed here. The 

* Sperry, 1918. 2, p. 214, 



38] TRANSFORMATIONS OF SURFACES 191 

differential equation of the union curves on the surface S x of the congruence 
xw generated by the line joining the point P x to the point (0, w%, w^ 1) can 
be written by the aid of the condition of united position 1^2^2+^3^3+^4 = 
and equations (43). Thus we obtain 

(49) X' = -p+Lws+(a-Lw*+2w3)\-(8+2w2-Nw3)\*+(y-Nw2)\* , 

wherein it is understood that X now stands for dv/du and X' for d?v/dvP. 
Similarly, the equation of the union curves on the surface S v of the congru- 
ence yz generated by the line joining the point P v to the point (0, z 2 , z s , 1) is 

(50) M '=-/3'+(a'+2z 3 )M-(5'+2z 2 V+7V, 

wherein /* = dv/du and n' = d 2 v/du 2 . If to every curve (49) corresponds a 
curve (50), and vice versa, these two equations must give X' = // whenever 
X = /i. Conditions necessary and sufficient therefor are 






If, further, none of the asymptotic curves correspond on the two surfaces, 
then LNy^Q and the congruence xw consists of the axes of the correspond- 
ence at the points of the surface S x , while the congruence yz consists of the 
axes of the correspondence at the points of the surface S v . But if the union 
curves on the surface S x of every congruence xw correspond to the union 
curves on the surface S y of some congruence 2/2, then the first two of equa- 
tions (51) are identities in w 9 , w Z) and we have the conditions 

(52) L = N = p-p' = y-y' = 0. 

Under these conditions not only do the asymptotic curves correspond on the 
surfaces S x and S y but these two surfaces are related so that they are, as we 
shall see in the next paragraph, protectively* applicable. 

Projective applicability of two surfaces was first defined and studied by 
Fubini. Later Cech provedf that Fubini's definition was equivalent to 
saying that two surfaces are protectively applicable in case their points can be 
placed in a one-to-one correspondence such that the bundle of planes at each 
point of one surface is projective with the bundle of planes at the corresponding 

* Fubini and Cech, 1926. 1, pp. 118-24. 
t Cech, 1922. 8. 



192 PROJECTIVE DIFFERENTIAL GEOMETRY 

point of the other surface, when corresponding planes are the osculating planes 
of corresponding curves at these points. Fubini's definition is also equivalent 
to the following characterization. Two surfaces are projectively applicable in 
case there is a one-to-one correspondence between their points such that* to all 
curves through a point on one surface with their osculating planes at this point 
forming a pencil correspond curves through the corresponding point on the 
other surface with their osculating planes at this point also forming a pencil. 
We arrived at this property at the close of the preceding paragraph. It 
follows that at corresponding points of corresponding curves on two pro- 
jectively applicable surfaces the differential invariant 



(see 24) is the same, just as in the theory of metrically applicable surfaces 
the differential invariant Edu z +2Fdudv+Gdv* representing the squared 
element of arc length is the same at corresponding points of corresponding 
curves (see 44, Chap. VI). Just as the first fundamental coefficients 
E, F, G at corresponding points of two metrically applicable surfaces are 
the same, so the coefficients /?, 7 at corresponding points of two projectively 
applicable surfaces are the same, and the projective differential invariant 
above is appropriately called the projective linear element. 

39. Quadratic nets and congruences, and the transformation of Ribau- 
cour. We begin by stating some definitions. A conjugate net in a space S n 
is called a quadratic net in case the net lies entirely on a non-singular hyper- 
quadric; similarly, a congruence of lines on a non-singular hyperquadric 
is called a quadratic congruence. Such a congruence should not be confused 
with an algebraic congruence of the second degree. Two quadratic nets on 
the same hyperquadric and in the relation of a fundamental transformation 
are said to correspond by a transformation of Ribaucour. This transforma- 
tion will be studied briefly in this section. 

The analytic criteria for a quadratic net now engage our attention. Let 
us consider a parametric quadratic net N x in a space S n . The coordi- 
nates Z of a variable point P x on the net N x not only satisfy a Laplace 
equation of the form (1) but also satisfy a homogeneous quadratic equation 
in n+1 variables. Since a non-singular quadratic form in n+1 variables 
can be reduced by a projective transformation to the sum of n+l squares, 
the latter equation can be written with customary abbreviation in the form 

(53) Sz 2 = 0. 

Consequently, equations (1), (53) represent analytically a quadratic net. 
* Fubini and Cech, 1926. 1, p. 122. 



39] TRANSFORMATIONS OF SURFACES 193 

We shall now prove that if a congruence is conjugate to a quadratic net and 
if each line of the congruence meets the hyperquadric on which the net lies in just 
two distinct points, then the second intersection of the congruence and the hyper- 
quadric is also a net conjugate to the congruence. These two nets are F trans- 
forms of each other, and such a transformation F is by definition a trans- 
formation* of Ribaucour. For the purpose of the proof let us consider a 
quadratic net N x with x satisfying equations (1), (53). Let us also consider 
a congruence ??f with focal nets AT,, Nf and conjugate to the net N x ; let rj 
satisfy the equations 

(54) T/U = (<pu+b<t>)x , TJ V = <p(x v - ax) , 

where p is a solution of the adjoint of equation (1), according to the con- 
clusion of Section 32. The point y defined by 

y = y + Xz (X scalar) 

is on the line xrj, and this point is also on the hyperquadric (53) in case 
Si/ 2 = 0. In view of the fact that x satisfies equation (53), we find that the 
line xt) meets the hyperquadric (53) in the point x, of course, and in the point 
y uniquely determined by solving for X the equation 

(55) F+XX = 0, 

in which the functions X, Y are defined by placing 

X = 22xrj, y = Sr/ 2 (XIVO) . 

f 
The intersection point y is therefore given by the formula 

(56) y = Xr,-Yx, 

the proportionality factor for y having been suitably chosen. It is not diffi- 
cult to show that X, Y satisfy equations (54) when substituted in place ofx, rj 
respectively. It follows that X is a solution of equation (1), while Y is a solu- 
tion of the Laplace equation satisfied by rj, namely, 



Consequently it is easy to verify the equation 



* Ribaucour, 1872. 1, p. 1491. 



194 PROJECTIVE DIFFERENTIAL GEOMETRY 

Comparison of this equation with equation (IV, 81) shows that the point y 
generates a net which is a Levy transform of the net Ni, and which is there- 
fore conjugate to the congruence xy, that is, the congruence r?f . This com- 
pletes the proof. 

By definition a congruence in space S n is quadratic in case all of its lines are 
on a non-singular hyperquadric. If a congruence is quadratic it follows at 
once that both of its focal nets are quadratic. But if a net is quadratic it 
does not necessarily follow that the congruence of tangents of the curves of 
one family of the net is quadratic. In fact, the congruence of v-tangents of a 
quadratic net (1), (53) is itself quadratic if, and only if, Sz? = 0. 

It can be shown that if one focal net of a quadratic congruence is subjected 
to a transformation of Ribaucour, this transformation carries the congruence 
into a quadratic congruence. It is sufficient to consider the quadratic net (1), 
(53) with Sz? = and to prove by direct calculation that y v derived from 
equation (56) is such that S?/? = 0. The details of the demonstration are 
omitted. 

It can further be shown that the two quadratic congruences xx v and yy v 
are harmonic to the net Nr,. It is sufficient to observe that equations (54) show 
that the line xx v is the same as the line rj u rj V) and to prove by a little calcula- 
tion that the line yy v is the same as the line joining the points (n/Y) u and 
(n/Y} v . Incidentally, it follows that corresponding lines xx v and yy v inter- 
sect in a point z which generates a net N z that is conjugate to both con- 
gruences (see Ex. 17 of Chap. IV). Obviously this net N z is itself quadratic. 

40. The correspondence between lines of R$ and points on a hyper- 
quadric in 5. We begin with a few comments on two symbols R$ and 
(aw). Ordinary projective space with the straight line as generating element will 
be called ordinary ruled space and will be denoted by the symbol R$, in spite of 
the fact that this space is four-dimensional. Recalling the definition of the 
pliickerian coordinates w of a line, as stated in Section 7, we define the symbol 
(aw) by the formula 

(57) (aw)=a 3 4Wi2 + a42Wi3-|-a23Wi44-ai4W23 + ai3W42-hai2W34 . 

Now the equation of a linear complex takes the simple form (aw) = 0, and the 
quadratic relation (I, 43) satisfied by the coordinates w can be written in 
the form (ww)=0. The linear complex is special, according to Section 20, 
in case (aa) = 0. 

The correspondence with which this section is concerned is defined as 
follows. Let the six homogeneous coordinates w of a line in ordinary ruled space 
Rsbe interpreted as the projective homogeneous coordinates of a point in a linear 
space S&; then to the lines of space R^ correspond the points on the hyperquadric 



40] TRANSFORMATIONS OF SURFACES 195 

Q 4 whose equation is (coco) = in the space $5, corresponding point and line 
having the same or proportional coordinates. This correspondence, which 
seems to have been pointed out* first by Klein, and to have been studied 
extensively f first by Segre, has proved to be a fruitful transformation of 
problems concerning surfaces and congruences in ordinary space. Some of 
the essential features of this correspondence will be summarized in this sec- 
tion so that they will be available for use in connection with the applications 
of the transformation in Sections 41 and 43. 

Two lines l u , 1 T with coordinates w, r in space R$ can be shown to intersect 
in case (cor)=0, and then any linear combination pco+gr givesf the co- 
ordinates of a line in the flat pencil determined by l a , 1 T . In space $5 the 
corresponding points P u , P r determine a straight line which lies entirely on 
the hyperquadric Q 4 . Hence we have the theorem: 

To a flat pencil of lines in ordinary ruled space R s corresponds a rectilinear 
generator of the hyperquadric Q 4 , each line of the pencil corresponding to a 
point of the generator. 

Three lines ? u , 1 T , 1 in space R s , not belonging to a flat pencil, intersect in 
pairs in case (cor) = (TO) = (<ro>) =0. In this case (see Fig. 30) the lines either 
belong to a ruled plane or to a bundle of lines. Then any linear combination 
pu+qr+ra gives the coordinates of a line intersecting all three lines l u , 1 T , 1 9 
and hence belonging to the ruled plane or to the bundle as the case may be. 
In space >S 5 the corresponding points P^, P T , P ff determine a plane which 
lies entirely on the hyperquadric Q 4 . It is known that there are two systems 
of planar generators on a hyperquadric in space S&, just as there are two 
systems of rectilinear generators on a quadric in space $3. We formulate 
our result as follows: 

To the ruled planes of space R$ correspond the planar generators of one family 
of the hyperquadric Q 4 , and to the bundles of lines of R s correspond the planar 
generators of the other family of Q 4 . 

To a linear complex with the equation (aco) = in space R s corresponds the 
intersection of the hyperquadric Q 4 by the hyperplane represented in space 
$5 by the same equation. The point a, which is the pole of this hyperplane 
with respect to Q 4 , is called the second image of the complex. The complex 
is special in case the point a is on the hyperquadric Q 4 . So we have the 
theorem : 

To the linear complexes of space R s correspond as second images the points 
of space S&, special linear complexes in # 3 corresponding to points on the hyper- 
quadric Q 4 . 

* Klein, 1872. 2, p. 261. t Segre, 1885. 1. 

t Bertini, 1923. 2, p. 158; Graustein, 1930. 1, pp. 463-65. 



196 



PROJECTIVE DIFFERENTIAL GEOMETRY 



The lines common to two complexes with the equations (aw) = 0, (6w) = 
in space R& form a linear congruence which corresponds to the intersection 
of the hyperquadric $4 by the space 83 represented in space S& by the same 
two equations, and which has for second image the polar line ab of the space 
3 with respect to Q 4 . There are three possibilities as to the intersection of 
the line ab and the hyperquadric Q*; accordingly, there are three classes of 
linear congruences in space RZ. First of all, the line ab may intersect the 





FIG. 30 

hyperquadric Q4 in just two distinct points PI, PZ. Every point on the line 
ab is the second image of one of a pencil of linear complexes containing the 
congruence. The two points PI, P% are the second images of two special 
linear complexes containing the congruence. Therefore the congruence has 
two distinct skew straight line directrices corresponding to the intersection 
points PI, P%, and consists of the straight lines intersecting these two skew 
lines. Secondly, in case the line ab is tangent to the hyperquadric Q*, the 
congruence has two coincident directrices. Finally, if the line ab lies entirely 
on Q 4 , all the complexes of the pencil containing the congruence are special, 



40] TRANSFORMATIONS OF SURFACES . 197 

their axes forming a flat pencil; the congruence consists of lines intersecting 
all the lines of this pencil. 

Conjugate lines with respect to a linear complex may be defined as follows. 
Let us consider in space R$ a non-special linear complex (aco) = and a line 
lb. This line may be regarded as the axis of a special linear complex (6w) = 0. 
Then the two complexes determine a pencil, and the other special complex 
in this pencil has for second image in space S 5 the second intersection P c of 
the line ab with the hyperquadric #4. In space R* the lines k and l c are said 
to be conjugate, or polar, with respect to the complex (aw) = 0. It may be 
shown that the lines of a complex which intersect one of two lines conjugate 
with respect to the complex intersect the other also, forming a linear con- 
gruence; and all lines intersecting both conjugate lines belong to the com- 
plex. 

The lines common to three complexes with the equations (ao>) = 0, (fra>) = 0, 
(cw) = in space #3 form a configuration which will be shown to be a regulus, 
that is, to consist of lines intersecting three skew lines. The configuration 
corresponds to the intersection of the hyperquadric Q 4 by the plane repre- 
sented in space 85 by the same three equations, and has for second image the 
polar plane abc of this plane with respect to Q 4 . Every point in the plane 
abc is the second image of one of a net, or two-parameter family, of com- 
plexes containing the configuration under discussion. Let us suppose that 
the points P a , Pb, P c are on the conic of intersection of the plane abc and Q 4 , 
none of the lines ab, be, ca being generators of Q 4 . Then every line J w of the 
configuration in space jR 3 meets the three lines l a , k, l c , and the configura- 
tion is therefore a regulus, as was to be shown. The lines linearly dependent 
on l a , lb, k form the other regulus on the quadric sustaining the first. The 
special relative positions of the plane abc and the hyperquadric Q 4 are in 
accord with the various types of degenerate reguli. From this discussion we 
have the following theorem: 

Two reguli on the same quadric in space R$ correspond to the conies of inter- 
section of the hyperquadric Q 4 by two planes that are reciprocal polar s with 
respect to Q 4 . 

To a ruled surface in space R 3 corresponds a curve on the hyperquadric Q 4 
in space $ 5 . If the ruled surface is developable, then any two consecutive 
generators intersect determining a pencil of lines, and the tangents of the 
curve on the hyperquadric Q 4 are generators of Q 4 . 

In conclusion we shall indicate synthetically an interesting application of 
the transformation studied in this section. To a non-ruled surface S in 
space .Ra, regarded as the envelope of the complex* of its tangents, corre- 

* Bompiani, 1912. 1, p. 406. 



198 PROJECTIVE DIFFERENTIAL GEOMETRY 

spends a variety F 3 on the hyperquadric Q 4 . But since the complex of 
tangents of the surface S is composed of < 2 pencils of lines, the variety F 3 is 
composed of 2 lines; this variety is, in fact, a quadratic congruence. To 
the congruence of lines tangent to the surface S at the points of a curve on 
S, which is composed of l flat pencils of lines, corresponds a ruled surface 
in the congruence F 3 . In particular, to the congruence of tangents of the 
surface S along an asymptotic curve on S, in which two consecutive pencils 
have a line in common, namely an asymptotic tangent, corresponds a de- 
velopable in the congruence Fa, on which two consecutive generators inter- 
sect in a focal point of a generator. To the developable of tangents of an 
asymptotic curve on the surface S corresponds the edge of regression of a 
developable in the congruence F 3 . To the congruence of tangents of one 
family of asymptotic curves on the surface S corresponds one focal surface 
of the congruence F 3 ; on this surface the edges of regression of one family 
of developables of F 3 represent the developables in the congruence of asymp- 
totic tangents of S, while the curves conjugate to these edges of regression, 
i.e., the curves of contact of the developables of the other family of the con- 
gruence Fs, represent the ruled surfaces in the congruence of asymptotic 
tangents of the surface S constructed at the points of the asymptotic curves 
of the other family. 

41. Surfaces in ordinary ruled space. The purpose of this section is to 
apply the transformation discussed in the preceding section to the theory 
of non-ruled surfaces in ordinary ruled space. The methods used are ana- 
lytic rather than synthetic as they were in the preliminary discussion of 
the problem at the close of the last section. The surface under consideration 
is referred to its asymptotic curves as in Chapter III, and by means of the 
transformation some of the theory of Chapter IV and the earlier sections 
of the present chapter, particularly the theory of quadratic nets and con- 
gruences and of reciprocally polar sequences of Laplace in space S n , can 
be used by placing n = 5 to deduce results concerning the geometry of 
surfaces in ordinary space. These results principally concern certain com- 
plexes considered by Wilczynski, the directrices of Wilczynski, the canonical 
pencils, the tetrahedron of Demoulin, the curves of Darboux, and those 
surfaces whose asymptotic curves belong to linear complexes. 

We proceed to calculate two Laplace equations which play a large part in 
this section. Let us consider in ordinary space' an integral surface S of 
equations (III, 6). The pliickerian line coordinates co of the u-tangent, and 
the coordinates r of the v-tangent, at a point P x on the surface S are pro- 
portional to certain determinants of the second order, which can be written 



41] TRANSFORMATIONS OF SURFACES 199 

with customary abbreviation in the respective forms (a:, x u ) and (z, &).. Let 
us choose the two proportionality factors for these coordinates so that 

(58) co = e~*(x, x u ) , T = e~ e (x, x v ) . 
Differentiation and reduction by means of equations (III, 6) give 

(59) W tt = 0T, T V = 70>, 

and then 

(60) = 07+ (log #)t,w tt , T UV = jfryr + (log y) u T v . 



These are the Laplace equations which we set out to calculate. Employing 
the transformation of the preceding section, we may formulate some imme- 
diate consequences as follows. 

The two asymptotic tangents Z w , 1 T at a point P z on a surface S in space Rs 
correspond to two points P u , P T on the hyperquadric Q 4 in space S&. As P x 
varies on S, the points P u , P T generate two quadratic nets N u , N T on Q 4 . The net 
NV is the first Laplace transform of the net N r . The nets N u , N T are the focal 
nets of the congruence V s which represents on Q 4 the complex of tangents of 
the surface S. 

The coordinates w satisfy not only the first of equations (60) but also a 
linear homogeneous partial differential equation of the third order which can 
be found in the following way. Making use of the first of equations (58) 
and the equations obtained therefrom by differentiation, and employing 
equations (6), . . . , (10) of Chapter III, we calculate the following formu- 



(61) 



+(log 

+(log 



, x uv ) 



Then differentiating the last of these equations with respect to v and re- 
ducing the result, we obtain the equation sought for, namely, the second 
equation of the following system, in which the first equation is merely the 
first of (60) rewritten for convenience of reference, and the functions I, m 
were defined in Section 16: 



( 62 ) 



200 PROJECTIVE DIFFERENTIAL GEOMETRY 

Performing the requisite calculations would show that the system (62) is 
completely integrable. This system is fundamental for the study of the net N u 
that corresponds on the hyperquadric Q 4 to the congruence of asymptotic 
w-tangents of the surface S in ordinary ruled space R*. A symmetric system 
of equations could be written for the net N T which corresponds to the con- 
gruence of asymptotic v-tangents of the surface S. 

Some further comments on the net N u may be based on the first of equa- 
tions (62). The invariants H, K of the net AT W are given by the formulas 

(63) H = /3 T ~ (log 0) u , , K = /3 7 . 

The first Laplace transformed net Ni of the net N u is generated by the point 
o)i defined by 

on =0^ (log j8) v w . 

By means of equations (IV, 19) the coefficients and invariants of the 
Laplace equation for the net Ni are found to have the values given by 

rai= (logoff)., 6i = 0, ci = H, 
(o4) < 

l#i = #-(lo 



In general, for the rth transformed net N r of the net N u there is a Laplace 
equation whose coefficients and invariants are shown by equations 
(IV, 23) to have the values given by 



(65) '.jr r -n^-aog 

Similar formulas could easily be written for the net N T . 

Let us consider a point P u of the net N a in space *S 5 , as shown in Figure 31. 
Associated with P u we have six hyperplanes determined by the following six 
sets of five points each: 

O), 0)1, 0)2, 0)3, 0)4 J 
T, 0), 0)1, 0) 2 , 0) 3 J 
T_i ,T ,0), 0)1, 0) 2 J 

r_ 2 ,r_i ,r ,0) ,0)1 ; 
r_4 ,r_ 3 ,r_ 2 ,r_i ,r . 

Each of these hyperplanes may be regarded as the osculating hyperplane 
of a v-curve on the surface generated by the first point mentioned in deter- 



41] 



TRANSFORMATIONS OF SURFACES 



201 



mining it, and also as the osculating hyperplane of a w-curve on the surface 
generated by the last point mentioned in determining it. Moreover, the 
third hyperplane is the space /S(2, 0), denned as in Section 27, at the point 
P w of the surface S u sustaining the net N^, while the fourth hyperplane 
bears the same relation to the net N T . It can be shown that the poles of these 
six hyperplanes with respect to the hyperquadric $4 are respectively the six 
points r_2, T_I, T, oj, coi, o> 2 . For this purpose it is sufficient, according to 
Section 33, to make the demonstration in one instance; let us choose the 





FIG. 31 



third for simplicity. The polar hyperplane of the point P T has the equation 
(r)=0 in variable coordinates , and it is not difficult to show that this 
equation is satisfied when each of T_I, T, w, o>i, o>2 is substituted in place of 
, thus completing the proof. 

The six hyperplanes just considered intersect the hyperquadric Q 4 in six 
varieties V$ which correspond to six linear complexes in space R^. The inter- 
section of Q 4 by the first hyperplane corresponds to the osculating linear 
complex* L u along the generator xx u of the ruled surface of u-tangents con- 
structed at the points of the v-curve through the point P x on the surface $, which 
therefore has for second image the point r_ 2 . Similarly, the osculating linear 
complex L v along the generator xx v of the ruled surface of t;-tangents con- 
structed at the points of the w-curve through the point P x on the surface S 
has for second image the point o>2. The osculating linear complexes M v and 

* Wilczynski, 1908. 2, p. 83. 



202 PROJECTIVE DIFFERENTIAL GEOMETRY 

M u at the point P x of the v-curve and the u-curve on the surface S have for 
second images the points T_I and o>i respectively, while the points r, w may 
be regarded as the second images of the special linear complexes whose axes 
are respectively the v-tangent and the u-tangent at the point P x of the surface S. 
We shall now find the points on the hyperquadric Q 4 which represent the 
directrices of Wilczynski through a point P x on the surface S. These are the 
points where the line r_i i, which is the second image of the congruence of 
intersection of the linear complexes M v , M U) intersects the hyperquadric Q*. 
Let us place 

._. ( At = e~ 9 (x, x uv ) , A 4 = e- e (x u , x v ) , 

(66) < 

I A 5 = e~ (x uv , z u ) , AQ = e~ e (x v , x uv ) . 

The points w, T, A$, . . . , A 6 represent on the hyperquadric Q* the six edges 
of the tetrahedron x, x u , x v , x uv associated with the point P x of the surface S. 
Adding and subtracting the first and second of equations (61) we obtain 



(67) 

and it follows that 



T_i = 8 w <pr 
(68) l 

< 



Now the point COI+T-I is on the hyperquadric Q^ since it lies in the planar 
generator cor^L 3 which represents the bundle of lines with center at P x . 
Therefore the point WI+T_I represents the first directrix of Wilczynski. Simi- 
larly, the point coi r_i lies on the planar generator curA 4 which represents 
the ruled tangent plane of S at P X) and therefore the point wi r_i repre- 
sents the second directrix of Wilczynski. Incidentally, we observe that the 
point Az represents the projective normal^ and therefore the line A 3 (coiH-r_i) 
represents the first canonical pencil, while the line A^I T-I) represents the 
second canonical pencil } at the point P x of the surface S. 

At this place we exhibit a formula which will be useful later. Let us con- 
sider in ordinary space a surface S generated by a point P x , and let us 
define two points Y, Z by placing 



Z =Zi 



41] TRANSFORMATIONS OF SURFACES 203 

The general coordinates 12 of the line YZ are found by direct calculation to 
be given by the formula 

(69) & = Wi2W + Wi3T+COi 4 A3+C023A4+W42^5+W34Ae , 

where o>, r are defined by equations (58); the symbols A^ . . . , A& are de- 
fined by equations (66); and the local coordinates ,* of the line YZ are 
defined by the usual formula u t k = y,?kykZi. The formula (69) gives the six 
general coordinates of a point ft on the hyperquadric Q 4 as linear combinations 
of tfie general coordinates of the vertices of the pyramid co, T, A 3 , . . . , A 6 , the 
coefficients * being the local coordinates of the point ft with reference to this 
pyramid. 

In order to bring the quadric of Lie into consideration again, we make the 
following remarks. The intersection of the hyperquadric Q 4 by the osculat- 
ing plane cocoico2, at a point P u of a v-curve of the net A r w , represents the 
osculating regulus along the generator xx u of the ruled surface of w-tangents 
constructed at the points of the 0-curve through a point P x on the surface S. 
Similarly, the intersection of Q 4 by the osculating plane Tr_ir_ 2 , at the cor- 
responding point P T of the w-curve of the net N T that passes through P T , rep- 
resents the osculating regulus along the generator xx v of the ruled surface of 
y-tangents constructed at the points of the w-curve through the point P x on 
S. These two planes are reciprocal polars with respect to the hyperquadric 
Q 4 , and the two reguli lie on the quadric of Lie at the point P x of the 
surface S. 

It was shown in Section 25 that as the point P x varies on the surface S 
the quadric of Lie touches its envelope, besides at P x , in four other points 
Piii Pi2, PM, PZI which are the vertices of the tetrahedron of Demoulin of S at 
P x . We shall now discover other properties of the edges of this tetrahedron, 
making use of their images on the hyperquadric Q 4 in space 85. On reference 
to the equation (III, 26) of the quadric of Lie it is easy to verify that any 
generator of the regulus thereon containing the u-tangent, x 3 = x 4 =0, has 
the equations 



in which h is a parameter. Furthermore, the two points 

(h, 0, 1, 0) , (0, 2V,l3y+d uv , 2/0 
are on this generator, and hence its line coordinates are 
[h 2 , (/3 7 +<U/2, h, -h, 0, 1] . 



204 PROJECTIVE DIFFERENTIAL GEOMETRY 

Therefore the point 12 corresponding to this generator on the hyperquadric 
Q 4 is given, according to the formula (69), by 

(70) V = h*u+(0y+d v )T/2+h(As-A*)+At . 
Eliminating T, A 3 A*, AQ, one is able to express 12 as a linear combination 

Of CO, COi, 602 1 

(71) 2l2 = [2/i 2 +2^-f(W) t ,/^-7^-2g]co+(2/i-haiH-^)a J iH-co 2 . 

The line coico 2 meets the hyperquadric Qt in two points which represent the 
two generators of the quadric of Lie which are the directrices of the con- 
gruence common to the complexes M v , L v . In order that the point 12 may 
coincide with one of these points it is necessary and sufficient that the co- 
efficient of co in equation (71) should vanish. Comparing the equation thus 
obtained with equation (III, 103) we see that* the edges PnPn, PzzPzi of the 
tetrahedron of Demoulin are the directrices of the congruence common to the 
complexes M u , L v , Similarly, the edges ^12^22, PziPn are the directrices of the 
congruence common to the complexes M v , L u . Since the edges P\\Pw, P\^P^\ 
intersect all the other edges it follows that the edges PuPzz, PizPzi are the 
two lines common to the four complexes L u , L v , M u , M v . Incidentally, it 
follows that the lines PnPzz, PizPzi intersect the directrices of Wilczynski. 

On the surface $ w in space $ 5 there are ordinarily just three one-parame- 
ter families of curves, called principal curves, each of which has the property 
that its osculating space 83 at any one of its points P u lies in the space 
S(2, 0) of S u at P w (see 54, Chap. VII). We proceed to consider these 
curves briefly. Their differential equation is found by demanding that 
each of co, co', co", to"' may be a linear combination of co, co u , u v , co uu , u vv . 
From this demand on co, co', co" no new conditions result, but from co'" we 
find, on making use of the second of equations (62), the desired differential 
equation, 

(72) ^u 3 +7^ 3 = 0. 

Consequently, the three families of principal curves on the surface $ repre- 
sent | the ruled surfaces of u-tangents circumscribed along the curves of Darboux 
on the surface S. The intersection of the hyperquadric Q 4 and the space 
S(2, 0) at a point P u of the surface >S W represents the special complex of 
lines intersecting the corresponding v-tangent of the surface S; the inter- 
section of Q 4 and the osculating space >S 3 at a point of a curve on the surface 

* Godeaux, 1927. 6. t Bompiani, 1926. 7, p. 399. 



42] TRANSFORMATIONS OF SURFACES 205 

Su represents the osculating linear congruence along the corresponding 
generator of the ruled surface of u-tangents circumscribed along a curve on 
the surface S. This congruence, which is determined by four consecutive 
generators of the ruled surface, consists of the lines intersecting the flec- 
node tangents of the generator. If the ruled surface is circumscribed along 
a curve of Darboux, the lines of the osculating linear congruence along a 
generator intersect the v-tangent that has the same contact point with the 
surface S as the generator. Hence we have the theorem of Cech (see Ex. 9 
of Chap. Ill) : 

Each of the two ruled surfaces of asymptotic tangents circumscribed along a 
curve of Darboux on a surface in space *S 8 has the other as a flecnode surface. 

We conclude this section with a few remarks concerning surfaces whose 
asymptotic curves belong to linear complexes. If the asymptotic w-curves on a 
surface S in ordinary space belong to linear complexes, then w satisfies an 
equation of the form SFo> = in which V is a function of v only. This is the 
equation of the complex M u whose second image is the point o>i. Therefore 
the locus of the point wi is a v-curve, and H 0. The first of equations (63) 
now shows that a necessary and sufficient condition that the asymptotic u- 
curves on a surface in ordinary space belong to linear complexes is (see Ex. 30 
of Chap. Ill) 

(73) /*y-(log0) w = 0. 

According to Section 28, the w-curves on the surface S T now lie in the same 
hyperplanes as the w-curves on the surface S a , and also lie in spaces $ 3 ; 
hence the ruled surfaces of v-tangents constructed at the points of the in- 
curves on the surface S belong to the same linear complexes as these in- 
curves and also belong to linear congruences. Moreover, the locus of the 
point r_3 is a 0-curve, the sequence of Laplace determined by the net N u 
terminating according to the case of Goursat in the negative direction, and 
according to the case of Laplace in the positive direction. Exercises 16, 17, 
18 give additional results connected with this section. 

42. Fubini's theory of W congruences. The concluding sections of this 
chapter will be devoted to W congruences, i.e., congruences on whose 
focal surfaces the asymptotic curves correspond. In this section Fubini's 
analytic theory of the projective differential geometry of these congruences 
will be outlined briefly. The reader who desires to see a more extensive 
exposition of this theory may consult* the treatise by Fubini and Cech. In 
the next section parts of this theory will be transformed by the correspond- 

* Fubini and Cech, 1926. 1, p. 243. 



206 PROJECTIVE DIFFERENTIAL GEOMETRY 

ence between lines in ordinary space and points on the hyperquadric Qt in 
space $ 6 . 

The first problem is to determine the second focal surface of the con- 
gruence of tangents of a one-parameter family of curves on a surface. Let us 
consider in ordinary space $3 an integral surface S x of equations (III, 6) 
wthout the restriction 0=log 0y characteristic of Fubini's canonical form of 
the differential equations. On S x let us consider a curve of the family de- 
fined by the equation 

Adv-Bdu = Q , 




FIG. 32 



wherein the coefficients A, B are functions of u, v (see Fig. 32). Any point 
P y on the tangent l xv at a point P x of this curve is defined by 



(74) y = vx+2(Ax u +Bx v ) (//scalar). 

As the point P x varies on a curve of the conjugate family 



the line l xv describes a developable surface. The point P y is the focal point 
of Zxy regarded as a generator of this developable, if, and only if, /x is given 
by the formula 

(75) M= -A u -B v -Ae u -Be v +A(B u +0A)/B+B(A v +yB)/A. 

When u, v vary independently, the locus of the point P y is a surface S y and 
the line l^ generates a congruence xy. One focal surface of this congruence 
is S x , and when ju is given by the formula (75) the other focal surface is S v . 
In order to find a condition necessary and sufficient that the congruence 
xy may be a W congruence, let us first demand that the w-curves on the 
focal surface S v shall be asymptotic curves. Then y must satisfy the 
equation (y UU) y f y u , y v )=0. By means of equations (74), (75) this equation 
can be reduced to 

(76) ((A V + 7 B)/A] U = ((B U +0A)/B] V ; 

the details of the reduction will be omitted. The symmetry of equation (76) 
shows that the v-curves on the surface S y are asymptotic if the u-curves are. In 



42] TRANSFORMATIONS OF SURFACES 207 

fact, equation (76) is a necessary and sufficient condition that the congruence 
xy may be a W congruence. 

Equation (76) can be simplified. In the first place this equation shows 
that there exists a function <p defined, except for a constant factor, by the 
differential equations 

(log ?). = (A v +yB)/A , (log ?) = (B U +0A)/B . 

If two functions A, B are defined by placing A = <pA, B = <pB, these defini- 
tions can be used to eliminate <p from its two defining differential equations. 
Thus we obtain 



If now the dashes are dropped, we have 

(77) A v =-iB, B=-f$A. 

These equations are equivalent to equation (76) under the choice of the 
proportionality factor <p, which has no geometrical significance. It is on 
these equations that Fubini's theory of W congruences rests. 

The next step is to calculate a system of differential equations for the 
surface S v . Let us rewrite equation (75), taking advantage of (77), and also 
let us define two functions X, N in the following equations: 

c = ~-A u -B v -A8 u -B8 v , \=-A u +B v -Ad u +B8 v , 

/'7Q\ J 



Then actual calculation gives 

(79) 

-py v -iry=N u x/2A 

where TT, x now have the values 



Eliminating x from the last three of equations (79) we reach a system of 
differential equations for the surface S v . This system is of the same form as 



208 PROJECTIVE DIFFERENTIAL GEOMETRY 

the system we are using for the surface S x ; its coefficients, indicated by 
dashes, are given by the following formulas: 



Cp = ir+\ 
iq = X -*N V 



) M , J8= -0-BN u /AN , 

q = X -*N V /2BN , 7= -y-AN v /BN , (? v = ^+(log AT), . 



If two functions S, T are defined by placing 

(81) N u = 2A/S , N v = - 
then it can be shown that S, T satisfy the equations 

(82) S, = 0T, T U = 7 S, 

whence it follows at once that the function N satisfies the equation of Laplace, 

(83) N uv +yBN u /A +pAN v /B = . 

The geometrical significance of the condition N = const, can readily be 
found. It will be observed, on inspection of the third of equations (79), 
that we are only interested in the case ]WO, since the locus S v is not a 
proper surface if N = Q. Moreover, equations (80) show that if N = const., 
then system (80) reduces to the system given in Exercise 15 of Chapter III 
for the coordinates of the tangent plane of the surface &.. In this case the 
tangent planes of the surface S x and the points P y are related by a correla- 
tion in which corresponding point and plane are in united position, and 
which is therefore a null system. Consequently we have the theorem: 

A W congruence with N = const, belongs to a linear complex. 

Let us consider a W congruence of which one focal surface is ruled and 
one not ruled. If the surface S v is ruled with the w-curves for generators, 
then j8 = and hence 0= -BN/AN. Equations (77), (83) may be used to 
show that 

(84) (log B/N) U = , (log B/0), = , (log N) uv = T ; 

from these equations it follows that equation (73) is satisfied. Thus we 
reach the following theorem. 

// a non-ruled surface S x is one focal surface of aW congruence of which the 
other focal surface S y is ruled, then the asymptotic curves on S x that correspond 
to the generators on S v belong to linear complexes. 

Let us finally consider a W congruence of which one focal surface is a 



43] TRANSFORMATIONS OF SURFACES 209 

quadric. If the surface S x is a quadric then = 7 = and by Exercise 4 of 
Chapter III it is possible to make p = q = 0, 6 = const. Then we find 




(85) #= U' Z -V'*-2UU"+2VV" , 

2VU"'/N , Jf = 2UV'"/N , 

where 7, V are functions of w alone and of v alone respectively and accents 
indicate differentiation. Therefore S v is a quadric surface in case U and V 
are polynomials at most of the second degree in u and v respectively with 
constant coefficients; in this case the congruence xy belongs to a linear com- 
plex, since N = const. If the surface S v is ruled but not a quadric, we may 
suppose 7 = 0, 07^0; still the ruled surface S v belongs to a linear congruence, 
since the equation /3 = has two solutions v = const. Finally, even if the 
surface S v is not ruled, all of the asymptotic curves on the surface S v belong 
to linear complexes, since 

(log 0) ww 



The reader is referred to Exercises 19, 20 for additional results. 

43. W congruences in space R$. The correspondence previously studied 
between lines of ordinary ruled space R 3 and points on a hyperquadric Q* 
in space $ 5 has been used* by Terracini to connect Fubini's theory of W 
congruences with the theory of conjugate nets, and in particular with the 
transformation of Ribaucour. Some of these connections will be explained 
in this section, in which the exposition will follow in its salient features the 
memoir of Terracini just cited. 

Darboux proved t that the line coordinates of a generator of aW congruence 
satisfy a linear partial differential equation of the second order. When the 
parameters are suitably chosen this is an equation of Laplace. We wish to 
compute this equation for the W congruence xy of the preceding section. 
For this purpose let us choose the proportionality factor for the coordinates 
t of a generator l xy of this congruence so that 



(86) t 

it being understood that the coefficients A, B satisfy equations (77) and 
w, T are defined by equations (58). Then, by means of equations (59), (60), 
(77), differentiation and elimination lead to the following equation of Laplace 
satisfied by t, 

(87) t uv = [07 - (log A) u (log B) v ]t+ (log B) v t u + (log A) u t v . 
* Terracini, 1927. 7. t Darboux, 1889. 1, p. 345. 



210 PROJECTIVE DIFFERENTIAL GEOMETRY 

For equation (87) the Laplace-Darboux invariants H, K are given by 
(88) H = fo- (log B) uv , K = ft- (log A) uv . 

It follows at once that H = K if, and only if, the developables of the W con- 
gruence xy touch the surface S x along the curves of an isothermally conjugate 
net. In this case, making A = B by a transformation of parameters, we 
obtain from (77) the condition ft, = y u ; thus we find that S x is an R surface, 
i.e., sustains an R net (see Ex. 22). 

If we define the coordinates co, f of the w-tangent and the v-tangent re- 
spectively at a point P v of the surface S v by the formulas 



(89) == e-'(y, y u ) , f = <r*(y, y v ) , 

where the function is defined, except for an additive constant, by 



then co, r satisfy equations of the same form as those satisfied by co, r but in- 
volving the dashed coefficients given by equations (80) in place of the co- 
efficients of equations (III, 6). Moreover, it is not difficult to establish the 
linear relation 

(90) AW+BT= -AU+BT . 

It was shown at the conclusion of Section 39 that when a focal net of a 
quadratic congruence is subjected to a transformation of Ribaucour, this 
transformation carries the congruence into another quadratic congruence. 
It was also shown that two corresponding lines of these congruences inter- 
sect in a point which generates a quadratic net conjugate to both congru- 
ences. This theory can be applied to the quadratic congruences COT and COT 
which represent on the hyperquadric Q* the focal surfaces S x and S v of a W 
congruence. In fact, we shall show in the next paragraph that the two nets 
Nu and Nz on the hyperquadric Q 4 are in the relation of a transformation of 
Ribaucour, and that so are the nets N r and N- T . Two corresponding lines of 
the congruences COT and COT are shown by equation (86) and the linear rela- 
tion (90) to intersect in the point t that represents the line l xy . This point 
generates a net N t on Q 4 , whose Laplace equation is (87), and which repre- 
sents the W congruence xy. 

Referring to Figure 33, we proceed to prove that the nets N u and Nz are in 



143] 



TRANSFORMATIONS OF SURFACES 



211 



the relation of a transformation of Ribaucour. In the latter part of Section 32 
we solved the problem of finding all congruences conjugate to a given net. 




FIG. 33 



Applying the theory there developed to the net N we write the adjoint of 
the first of equations (60) in the form 



(91) 



v = Lfry- (log j8)J^- (log ft)i<p u . 



The focal points 17, ? of a generator of the most general congruence conjugate 
to the net N w are determined from the equations 



(92) 



, 11 V = <p\U v (lOg ),,] , 



212 PROJECTIVE DIFFERENTIAL GEOMETRY 

where the function <p is a solution of equation (91). Now it happens that 
S/P is a particular solution of equation (91). With this solution equations 

(92) give 

(93) * = #(+S)/2B+<S/0 , f = N(a+u)/2B . 

Since the line TJCO passes through the point S3, the nets N u , Nz, are both con- 
jugate to the congruence tf and hence are in the relation of a transformation 
of Ribaucour, as was to be proved. The proof for the nets N T , N 7 can be 
made similarly; the first and second focal points, analogous to the points 
?7, f are the points ??', f ' given by 

(94) n' = tf(T-T)/2A , ? = N(r-T)/2A + TT/y ; 

the solution of the adjoint of the second of equations (60) used is T/y. It 
will be observed that the points f, rf coincide in virtue of the relation (90), 
and that the points f ', f_i coincide. 

We shall next show how to prove that the net Nt representing the W con- 
gruence xy is the most general net conjugate to the congruence cor. The problem 
of determining all nets conjugate to a given congruence was solved in Sec- 
tion 32. Using equations (59) in place of (IV, 69) the reader may prove 
that equations (IV, 71), (IV, 72), (IV, 74) give for the most general net 
conjugate to the congruence O>T precisely the net N t of equation (87). 

Another application of the theory in Section 32 for the determination 
of all congruences conjugate to a given net will show that the congruence 
537 is conjugate to the net N t . The adjoint of equation (87) is 

(95) <p uv = [07- (log A) u (log )- (log AB) uv ]<p- (log B) v <p u - (log A) u <p v . 

The focal points r/', f " of a generator of the most general congruence con- 
jugate to the net N t are determined from the equations 



._. 

(96) 



where the function <p is a solution of equation (95). One solution of equa- 
tion (95) is N/AB; with this solution equations (96) give 

(97) Brj"=-Na , Bf"=-N 



43] TRANSFORMATIONS OF SURFACES 213 

The demonstration is completed by observing that this line ij"f " coincides 
with the line cor. Incidentally, another solution of equation (95) is 1/AB; 
with this solution equations (96) give 

(98) 77" = co, B$" = u-t/A. 

This line ?"f" coincides with the line COT; consequently the congruence thus 
determined as conjugate to the net N t is the congruence cor. 



EXERCISES 

1. If the product of two transformations F is a transformation F the three nets 
involved either are conjugate to the same congruence, or else are harmonic to the 
same congruence. (In the first case the three nets are said to form a conjugate triad, 
and in the second, a harmonic triad.) SLOTNICK, 1928. 3, p. 202 

2. Prove the theorem of permutability for transformations F, namely, that if two 
nets Ni, Nt are both F transforms of a net Nj then there exist < 2 nets N' such that 
each N' is an F transform of both Ni and AT 2 . Prove that four corresponding points 
on four nets Ni, Nz, N, N' lie in a plane which envelops a net, and that the tangent 
planes at four corresponding points of the nets meet in a point which generates a net. 

EISENHABT, 1923. 3, p. 45 

3. Using the points x, x u , x v , y in the notation of system (16) as the vertices of a 
local tetrahedron of reference, show that the equation of the quadric having con- 
tact of the second order with the surface S x at a point P x and contact of the first 
order with S v at the corresponding point P v is 



Show also that the equation of the quadric defined with the roles of x and y inter- 
changed is 



Discuss the cones contained in the pencil determined by these quadrics. 

LANE, 1929. 1, p. 466 

4. In the coordinate system of Exercise 3, the equations of the ray of the point 
P*are 



and those of the ray of P v are 



Hence the cross ratio of the second and first focal points x u , x v of the generator of 
the harmonic congruence and the points p, a where this line meets the rays of P x 



214 PROJECTIVE DIFFERENTIAL GEOMETRY 

and P y is the conjugate invariant R. Dually, the equation of the plane containing 
the line xy and the axis of the point P x is 



and the equation of the plane containing this line and the axis of P y is 



Hence the cross ratio of the second and first focal planes (x^ = and x 2 = 0) of the 
line xy and these two planes is the harmonic invariant S. 

LANE, 1929. 1, p. 464 

5. If a net N x with equal invariants is conjugate to a congruence with focal nets 
Ni,, Nf, the point which is the harmonic conjugate of a point P x with respect to the 
corresponding points Pr,, Pf generates a net with equal invariants which is conjugate 
to the congruence. 

6. If a net N x in space S n has equal invariants there exists a conjugate congruence 
with focal nets Nq, Nf such that the net generated by the point which is the harmonic 
conjugate of P x with respect to Pr,, Pf is in a hyperplane S n -\. 

7. The conic of Koenigs for a point of a net N x with equal invariants and the 
conic of Koenigs for the corresponding point of a net N v in the relation of a trans- 
formation of Koenigs to N x intersect the corresponding generator of the harmonic 
congruence of the transformation in the same two points. 

8. The quadrics of the pencil (26) intersect the line xy in pairs of points in an 
involution whose double points are the focal points Pr,, Pf ; the quadrics of the pencil 
that are tangent to the line xy are the cones of the pencil, the points of contact being 
the focal points Pr, } Pf . 

9. Find the vertices of the cones in the pencil (26). If the two conies determining 
this pencil are the conies of Koenigs, according to Exercise 7, the vertices are the 
points TJI, f_i given by equations (14). 

10. In the situation discussed in Section 35 any line through a point P x of a net 
NX and lying in the tangent plane of N x at P x intersects the conies of Koenigs (21) 
in four harmonic points if, and only if, H-}-K = 0. 

11. The cross ratio of the four points x-i, f, y-i, f_i of Section 34 is bn v /nK, 
and that of the points Xi, TJ, y\, -rji is am u /mH. 

12. In the situation of Section 37 the differential equations of the asymptotic 
curves on the surfaces S x , S v are respectively 

Ldu*+2M dudv+Ndv* = Q , L'du z +2M'dudv-{-N'dv*=Q . 

Use equation (40) to prove that the developables of the congruence pa correspond 
to a conjugate net on S x (supposed non-developable) if, and only if, the develop- 



TRANSFORMATIONS OF SURFACES 215 

ables of the congruence xy (supposed determinate and given by dudv = 0) intersect 
the surface S v in a conjugate net (Af' = 0). State the result when x and y are inter- 
changed. GROVE, 1928. 2, p. 487 

13. Necessary and sufficient conditions that the developables of the congruence 
pa be determinate and correspond to the developables of the congruence xy are found 
from equation (40) to be 

mLM'-nML' = mMN'-nNM' = Q , m(LN'+MM')-n(NL'+MM')y*Q . 

These conditions are satisfied if the developables of the congruence xy cut both 
surfaces S x , S v in conjugate nets (Af = M' = 0), or else if these developables cut both 
S x , S v in their asymptotic curves (L = AT = L' = JV' = 0) . Ruling out these two cases 
prove that 



Hence prove that at a point P x the cross ratio of the two tangents of the curves 
dudv = 0, in which the developables of the congruence xy intersect S x , and the two 
asymptotic tangents is equal to the cross ratio (in the proper order) of the two 
tangents at the corresponding point P v of the curves dudv = Q, in which the develop- 
ables of the congruence xy intersect S v , and the two asymptotic tangents at P v . 

GROVE, 1928. 2, p. 487 

14. Necessary and sufficient conditions that the foci of each line p<r be deter- 
minate and lie on the parametric tangents at the corresponding point P x are found 
from equation (41) to be 

MN' - NM f = MU - LM' = , m(LN f - MM') - n(NL' - MM') * . 

These conditions are satisfied if the developables of the congruence xy cut both 
surfaces S x , Sv in conjugate nets (M = Af' = 0), or else in their asymptotic curves 
(L = N=L'=?N' = Q), or else if the asymptotics correspond on the surfaces S x , 
S v ((L/L' = M/M' = N/N f ). Ruling out these three cases prove that 



so that the cross ratio property of Exercise 13 is present. 

GROVE, 1928. 2, p. 487 

15. Every quadratic net with equal invariants in ordinary space is an integral 
net of a system of equations which can be written in the form 



where the coefficients c, d are defined by 



the functions *>, t being arbitrary functions of the arguments indicated. 

TZITZEICA, 1924. 3, Chap. IV 



216 PROJECTIVE DIFFERENTIAL GEOMETRY 

16. The planes a>iw 2 w3 and T-ir_2T- 3 in the situation of Section 41 are reciprocal 
polars with respect to the hyperquadric Q 4 , and therefore intersect Q 4 in conies repre- 
senting reguli on a quadric surface. This quadric contains the four edges of the 
tetrahedron of Demoulin that lie on the quadric of Lie at a point x of the surface S; 
hence the two quadrics are tangent at the vertices of this tetrahedron. 

GODEAUX, 1927. 6 

17. For each point w on the surface S u of Section 41, the point 



lies on the hyperquadric Q\ and represents a generator of the regulus containing the 
w-tangent on the quadric (III, 86) at a point x on the surface S. As the generator 
varies over the regulus, the locus of this point is the conic of intersection of Q 4 and 
the plane determined by the three points 



This plane lies in the space S(2, 0) at the point w of the surface $ w . The regulus 
containing the ^-tangent on the asymptotic osculating quadric (III, 29) of a curve 
at the point x on the surface *S corresponds to the intersection of the hyperquadric 
Q 4 and the osculating plane at the point w of a curve on S u . For a planar system 
of curves as defined in Section 22, these two planes intersect in a straight line. 

BOMPIANI, 1926. 2 

18. The osculating plane at a point co of a principal curve on the surface S u and 
the osculating plane at this point of the harmonic reflection of this principal curve 
in the parametric curves, which represents the ruled surface of w-tangents touching 
the surface S along a curve of Segre, intersect in a straight line. The three principal 
curves at the point w thus determine three straight lines which lie in the plane of the 
three points 

w , 3co M u (log 0/7) uwu , 3 to (log b y) v a) V . 

19. If both focal surfaces S x , S v of a W congruence are ruled, and if to the curved 
asymptotics on S x correspond generators on S y , then S v is a quadric and the genera- 
tors of the second family on S v correspond to the generators on S x . 

SEGRE, 1913. 1 

20. If the curves of Darboux correspond on the two focal surfaces of a congruence, 
the congruence is a W congruence and both surfaces have the property of being iso- 
thermally asymptotic. (For an integral surface of equations (III, 6) this means that 
(log &/y)uv = 0.) Then it is possible to make = 7 by a transformation of parameters. 

FUBINI and CECH, 1926. 1, p. 283 

21. The cross ratio of the points w, w and the focal points TJ, f given by equations 
(93) is /3//3; the cross ratio of the points T, r and the focal points 17', f' given by 



TRANSFORMATIONS OF SURFACES 217 

equations (94) is -yfi (see Ex. 11). Therefore the harmonicness of these two sets 
of points implies that the curves of Darboux correspond on the two focal surfaces 
S x , S v of the W congruence xy in space $3. 

22. If in Section 42 the congruence xy consists of the tangents to the curves 
dv+\du = Q on the surface S x , so that B/A = X, the condition that the congruence 
be a W congruence becomes 

(log A).+(7A)-(0/X), = . 

If also the tangents to the conjugate curves dv\du=Q form a W congruence, the 
conjugate net dv 2 \ 2 du? = is isothermally conjugate and the net is an R net. Then 
by a transformation of parameters it is possible to make X = 1 and p v = T (see Ex. 15 
of Chap. IV). 

23. For the point f given by the second of equations (93) prove that 

r tt =-Sr, r,= -r, 

and hence derive the equation 



Of this equation a particular solution is the function N defined in equations (78). 
The Levy transform of the net Nf in the v-direction by means of this solution is the 
net Ns while the Levy transform of the net Nt in the same direction by means of a 
constant solution is the net N u . TEBBACINI, 1927. 7 

24. The point T in Exercise 23 is the pole, with respect to the hyperquadric Q 4 , 
of the space S* whose intersection with Q 4 represents the osculating linear complex 
along the generator l^ of the W congruence xy in space 83. TERRACINI, 1927. 7 

25. In the notation of Exercises 3, 4 the lines xx u , yx v , ijp, f a lie on the quadric 

axiX 3 bmx2Xi = ; 

and the lines xx v , yx u , ij<r, f p lie on the quadric 



LANE, 1929. 1, p. 470. This result is due to Mendel. 

26. In the situation discussed in Section 37 consider any point P not a focal 
point on any line Ixy of the congruence xy. Construct the two focal planes of Ixy and 
the two tangent planes at P of the two ruled surfaces of the congruence that con- 
tain Ixy and intersect the surfaces S x , S v in parametric curves thereon. Prove that 
the conditions w=n, 8^0 are necessary and sufficient that the first two planes 
separate the second two harmonically, whatever be the position of P on l^. (In 



218 PROJECTIVE DIFFERENTIAL GEOMETRY 

this case the parametric net of ruled surfaces of the congruence, composed of the 
ruled surfaces for which u= const, and those for which v= const., is said to be a con- 
jugate net of ruled surfaces.) WILCZYNSKI, 1920. 3, p. 203 

27. Calculate the integrability conditions of equations (42). Determine the 
transformations that leave the parametric net invariant, and calculate the effect of 
this transformation on the coefficients of the equations. GREEN, 1920. 2 

28. Consider a net N x with x satisfying equation (1) and consider a different net 
N v with y likewise satisfying (1). If the point z defined by an expression of the form 
z = x+hy describes a net when u } v vary, so that z also satisfies a Laplace equation 
with independent variables u, v, then this equation is identical with (1) and h= 
const. The surfaces sustaining four such nets N z intersect a line xy in four points 
whose cross ratio is the same for all lines xy. 



CHAPTER VI 
METRIC AND AFFINE APPLICATIONS 

Introduction. The aim of this chapter is to connect the projective differ- 
ential geometry of surfaces, as developed in the earlier chapters of this 
book, with the metric and affine differential geometries of surfaces in ordi- 
nary space. It is well known that the group of projective transformations 
in ordinary space contains as a subgroup the group of rigid motions. There- 
fore every invariant under the group of projections is also an invariant 
under the group of motions. The projective geometry of a configuration is 
the study of the properties of the configuration that are invariant under 
the projective group, and the metric geometry of the configuration is the 
study of the properties of the configuration that are invariant under the 
group of motions, which may be called the metric group. Therefore all of 
projective geometry may properly be included in metric geometry; every 
projective theorem is also a metric theorem, and every projectively defined 
configuration has metric properties. It is a matter of history that many 
theorems of a projective nature were discovered and included in metric 
geometry before projective geometry was organized as a separate science. 
The metric investigation of configurations that have appeared first in pro- 
jective geometry has been very fruitful as a general method of research, and 
is to be thought of in contrast with a second method, namely, the attempt to 
find projective analogues of familiar configurations and theorems of a 
metric nature. 

The general projective point coordinate system which is used in analytic 
projective geometry of ordinary space can be specialized so as to become an 
ordinary orthogonal cartesian coordinate system with equal units on all 
three axes. This specialization is accomplished by projecting one face of 
the tetrahedron of reference into the plane at infinity, by projecting the 
other three faces into three mutually orthogonal planes, and by suitably 
disposing of the unit point. Except in investigations involving elements at 
infinity it is customary to use non-homogeneous coordinates in metric ana- 
lytic geometry. 

A projective transformation that makes finite points correspond to finite 
points and makes points at infinity correspond to points at infinity is called 
an affine transformation. The group of affine transformations is therefore 
a subgroup of the projective group; moreover, the affine group contains 
the metric group as a subgroup. 

219 



220 PROJECTIVE DIFFERENTIAL GEOMETRY 

The contents of this chapter may be outlined as follows. In the opening 
section a few of the fundamental facts and formulas of the metric differential 
geometry of surfaces in ordinary space are collected for convenience of 
reference. In Section 45 some of the elementary portions of the geometry 
of the sphere are reviewed and pentaspherical coordinates are introduced 
to prepare the way for the definition and a preliminary account, in the fol- 
lowing section, of a certain correspondence between points in ordinary 
metric space and points on a hyperquadric in a projective space $ 4 . This 
correspondence is applied in Section 47 in studying the lines of curvature on 
a surface in ordinary space. 

In Section 48 a local trihedron at a point of a surface is used as the basis 
of a local cartesian coordinate system for the purpose of investigating 
metric properties of protectively defined configurations associated with 
the point. The connection between Fubini's normal coordinates, which were 
employed so extensively in Chapter III, and cartesian orthogonal coordi- 
nates is established in Section 49. 

Only the last two sections are concerned with affine geometry, and even 
that is of a very restricted kind. Section 50 is devoted to the elements of 
the special kind of affine geometry of surfaces which leaves the origin in- 
variant. Finally in Section 51 a certain projectively defined class of sur- 
faces, namely, surfaces with indeterminate directrix curves, is studied, first 
from the projective point of view and then from the point of view of the 
special affine geometry. 

44. Metric geometry of surfaces. It will be convenient for later use to 
collect here some of the definitions, formulas, and theorems of the metric 
differential geometry of surfaces in ordinary space. The notation employed 
will not diverge far from that of the treatise by Bianchi and of Eisenhart's 
Differential Geometry. These works may be consulted* by any one wishing 
to read a more extended account of the subject than the rather sketchy 
summary offered here. 

Let us establish a left-handed orthogonal cartesian coordinate system 
with equal units on all three axes in ordinary metric space. If the coordi- 
nates x } y, z of a point P in this space are single-valued analytic functions 
of two independent variables w, v, such that not all of the jacobians A, J5, C 
defined by the formulas 

(1) A = y^ v - y v z u , B = z u x v - z v x u , C = x u y v - x v y v 

* Eisenhart, 1909. 1; Bianchi, 1922. 5. 



44] METRIC AND AFFINE APPLICATIONS 221 

are zero, the locus of the point P, as u, v vary, is by definition a proper 
analytic surface S. The tangent plane at the point P of the surface S has the 
equation 

(2) Aft-*)+B(i-y)+C(r-*) = 0, 

in current coordinates , rj, f . 

TVie square of the element of arc of a curve on a surface is given by 

(3) ds 2 = dx 2 +dy 2 +dz 2 = Edu 2 +2Fdudv+Gdv 2 , 
where E, F, G are defined by the formulas 

(4) E = 2xl, F 



the summation being for cyclical permutation of x, y, z. The binary quad- 
ratic differential form in the last member of (3) is called the first fundamental 
form, and E, F, G are spoken of as the first fundamental coefficients, of the 
surface. Denoting by H 2 the discriminant EGF 2 of this form we have the 
equalities 

(5) H 2 = EG-F 2 = A 2 +B 2 +C 2 . 

Then, if X, Y, Z denote the direction cosines of the normal at a point of the 
surface, we find from equation (2) the formulas 



(6) X=A/H , Y = B/H , Z = C/H . 


The equation of the minimal curves, or curves of length zero, on the surface is 

(7) Edu 2 +2Fdudv+Gdv 2 = . 

If two surfaces have their points in one-to-one correspondence so that 
the length of any curve on one surface, calculated between any two points 
on it, is equal to the length of the corresponding curve between the corre- 
sponding points on the other surface, the surfaces are said to be applicable 
and the correspondence is called an applicability. If the parameters on two 
surfaces with their points in correspondence are chosen so that correspond- 
ing points have the same curvilinear coordinates, it can be shown that a 
necessary and sufficient condition that the correspondence be an applicability 
is that the first fundamental coefficients of one surface be respectively equal to 
those of the other surface. The surfaces then have the same first fundamental 
form. 



222 PROJECTIVE DIFFERENTIAL GEOMETRY 

The equation of the asymptotic curves on a surface is 

(8) Ddtf+Wdudv+D' 'dv* = , 
where the coefficients D, D', D" are defined by placing 

(9) D 



The binary quadratic differential form in equation (8) is called the second 
fundamental form, and its coefficients are spoken of as the second fundamental 
coefficients, of the surface. 

The Christoffel symbols {ik, 1} of the second kind for the first fundamental 
form of a surface may be defined by the formulas 



2# 2 }11, 1} = GE U +FE V -2FF U , 2# 2 )22, 2} = E 

2# 2 {21, 1} =GE V -FG U , 2# 2 {12, 2) =EG U -FE V , 

2# 2 {22, 1} = -FG V -GG U +2GF V , 2# 2 {11, 2} = -FE U -EE V +2EF U 



(10) 

The differential equations 



(11) 



x vv = \22,l}x u +{22,2}x v +D"X, 

X u = (FD f - GD)x u /H*+ (FD - ED')x v /H* , 

X v = (FD"-GD')x u /H*+(FD'-ED")x v /H* 



are shown in treatises on metric differential geometry to be satisfied not 
only by x, X but also by y, Y and z, Z. These are the fundamental differential 
equations in the metric differential geometry of surfaces in ordinary space. 

There are three integrability conditions of equations (11). One of these is 
the equation of Gauss, 

(12) HK=(H(ll,2}/E).-(H[12,2}/E)*, 

where K is the total curvature which will be defined in the first of equations 
(15). The other two are the equations of Codazzi, 



( 



rZ).-Di-{21,l}D+({ll,l}-{12,2 

lZ>i'-DJ-{12,2JZ)"+({22, 2} -{21, 



44] METRIC AND AFFINE APPLICATIONS 223 

The lines of curvature on a surface may be defined analytically by setting 
equal to zero the jacobian of the first and second fundamental forms of the 
surface. Then the curvilinear differential equation of the lines of curvature 
on a surface is 

(14) (ED' - FD)du*+(ED" - GD)dudv+(FD" - GD')dv* = . 

The lines of curvature thus defined analytically have the following char- 
acteristic geometric property. At each point P of a surface S the curve of 
section of S by a plane through the normal has a radius of curvature which 
is a minimum if the plane contains the tangent of one line of curvature 
through P, and a maximum if it contains the other. These extreme radii are 
called principal radii of normal curvature and are denoted in this book by 
Ri, R 2 . 

The total curvature K and the mean curvature K m at a point of a surface 
may be defined by the formulas 

K = l/RJtt=(DD"-D")/H* , 



If K = Q at every point of a surface, the surface can be proved to be a de- 
velopable. On the other hand, if K m at every point of a surface, the sur- 
face is called a minimal surface. 

The parametric curves on a surface can be shown to form an orthogonal 
net in case F 0. Similarly, the parametric curves can be shown to form a 
conjugate <net in case Z)' = 0. Equation (14) shows that the parametric 
curves are the lines of curvature (supposed determinate) in case F = D' = Q. 
It is known that the lines of curvature on a surface are indeterminate if the 
surface is a sphere or a plane. Thus one obtains the following characteriza- 
tion of the lines of curvature: 

On an unspecialized surface in ordinary space the lines of curvature are the 
only orthogonal conjugate net. 

If the lines of curvature are the parametric curves on a surface, then 
F = D' = Q and consequently the second of equations (11) becomes 



(16) x uv 

where the coefficients a, b are given by the simple formulas 
2a=(log) 1 ,, 26 = (logG%. 



224 PROJECTTVE DIFFERENTIAL GEOMETRY 

In this case it is easy to verify that the function w defined by 
(17) 2o> 



is also a solution of equation (16). Conversely, since the homogeneous 
cartesian coordinates x, y, z, 1 are a special kind of projective homogeneous 
point coordinates, it follows that if these four coordinates satisfy an equa- 
tion of Laplace 

(18) x uv = cx+ax u +bx v , 

then the parametric curves form a conjugate net, and c = 0. Indeed, it is 
easy to verify by direct calculation that D' = 0. Moreover, if the function w 
also satisfies the same equation (18), then F = and the parametric curves 
form an orthogonal net. Thus the following theorem is demonstrated : 

A necessary and sufficient condition that the parametric curves on a surface 
in ordinary space be the lines of curvature is that x,y,z,l, u satisfy an equation 
of Laplace of the general form (18) or what is equivalent, thatx, y, z, w satisfy 
an equation of Laplace of the particular form (16). 

46. Spheres and pentaspherical'coordinates. After a review of the ele- 
ments of the geometry of the sphere in the early part of this section, penta- 
spherical point coordinates will be defined, and some of the simpler facts' 
of sphere geometry will be expressed in terms of these coordinates, which 
were first used by Darboux. 

Let us recall the definition of a sphere and the criteria for the different 
types of spheres. In ordinary metric space the locus of a point whose car- 
tesian coordinates x, y, z satisfy an equation of the form 

(19) a(z 2 +2/ 2 +2 2 )+&z+q/+d2+e = , 

in which the coefficients are constants not all zero, is called a sphere. If 
a 7^0, the sphere is a proper sphere; the coordinates of its center and the 
length R of its radius are given by the formulas 

(-6/2o, -c/2o, -d/2o), # 2 =(& 2 +c 2 +^-4ae)/4a 2 . 

If a 7*0 and R = Q, the sphere is called a null sphere, or point-sphere; such a 
sphere may be regarded as a single point. If a = 0, the sphere is composite, 
one component being the plane 



45] METRIC AND AFFINE APPLICATIONS 225 



and the other being the plane at infinity. If #?*0, and if a approaches zero, 
then R approaches infinity. Hence a plane is sometimes spoken of as a 
sphere with an infinite radius, the plane at infinity not being then con- 
sidered as a component of the sphere. 

The power of a point P with respect to a proper sphere is defined to be the 
product of the distances from P to any two points on the sphere collinear 
with P. Hence the power of the point P is equal, by elementary geometry, 
to the square of the length of a tangent from P to the sphere. The power of 
a point (2, y, z) with respect to the sphere (19) may be shown to be expressed 
by the formula 

x*+y*+z*+(bx+cy+dz+e)/a . 

Two points are said to be inverse to each other with respect to a proper 
sphere in case* they are collinear with the center of the sphere and the 
product of their distances from the center is equal to the square of the radius 
of the sphere. The polar plane of each of two inverse points can be shown 
to pass through the other; therefore two inverse points are conjugate with 
respect to the sphere. In fact, any two conjugate points collinear with the 
center of a sphere are inverse to each other with respect to the sphere. 

Pentaspherical point coordinates are defined as follows. Let us consider 
five spheres with equations of the form (19), the determinant of whose co- 
efficients is not zero; and let us define the ratios of five numbers x\, . . . , x 5 
by placing 

(20) px^a^+yt+z^+bjX+Cjy+djZ+e,- (j=l, . . . , 5) , 



where p is* a proportionality factor not zero, and x, y, z are the cartesian 
coordinates of a point P. Then xi, . . . , x$ are by definition pentaspherical 
coordinates^ of the point P. The reader should observe how the pentaspheri- 
cal coordinates are connected with the powers of P with respect to the five 
spheres. It is clear that a different choice of the five spheres would result 
in a different pentaspherical coordinate system. 

It is possible to attain a desirable degree of analytic simplicity by choos- 
ing five mutually orthogonal spheres as the fundamental spheres. We shall 
use hereinafter the special pentaspherical coordinates xi, . . . , x b whose 
ratios are defined by placing 

(21) ( P l X ' P * ' P 8 ' 

lp&4 = i(3 2 +!/ 2 +3 2 +l)/2, p 5 = (z 2 4-^ 2 +z 2 

* Coolidge, 1916. 2, p. 227. f Darboux, 1887. 1, p. 213. 



226 



PROTECTIVE DIFFERENTIAL GEOMETRY 



where i 2 = 1 and p is a proportionality factor not zero. It will be observed 
that the fundamental spheres for this system of pentaspherical coordinates 
are the three coordinate planes of the cartesian coordinate system regarded 
as spheres of infinite radii, the unit sphere z 2 +2/ 2 -fz 2 1=0, and the im- 
aginary unit sphere z 2 -f2/ 2 +2 2 +l=0, which are all mutually orthogonal. 
Moreover, direct calculation shows that the sum of the squares of these 
coordinates x\, . . . , x& is zero, so that the special pentaspherical coordinates 
x of a point satisfy the quadratic equation 

(22) S* 2 = 0, 

the summation here, as elsewhere in this section, ranging over the integers 
1, . . . , 5. 

Coordinates of a sphere may be defined as follows. A linear homogeneous 
equation 



conveniently abbreviated into Sax = 0, in pentaspherical coordinates x and 
with constant coefficients a not all zero, represents a sphere whose center in 
cartesian coordinates, and whose radius R, can be shown by direct methods 
to be given by the formulas 



(23) 




The five coefficients a are called co- 
ordinates of the sphere, and one 
speaks of this sphere as the sphere 
a. The sphere is a point-sphere in 
case 2a 2 = 0, p^O; and is a plane in 
case p = 0. 

The angle 6 between two spheres 
a, b with centers (a!, a 2 , a s ), (ft, 
02, ft) and radii R, R' can be cal- 
culated by means of the equation 
from the elementary geometry of a 
triangle, as in Figure 34, 



FIG. 34 



By the aid of the formulas (23) this equation can be reduced to 

(24) Sa6 



46] METRIC AND AFFINE APPLICATIONS 227 

Therefore a necessary and sufficient condition that two spheres a, b be orthog- 
onal is 

(25) Sa6 = 0. 

The condition for tangency of two spheres a, b is 
(26) 



It is now possible to deduce* from the concluding theorem of the last 
section a very significant theorem. Let x, y, z be the non-homogeneous car- 
tesian coordinates of a point P describing an unspecialized surface referred 
to its lines of curvature in ordinary metric space. Since x, y, z, 1, co satisfy 
an equation of Laplace, it follows that the pentaspherical coordinates 
xi, . . . , x b of the point P, whatever be their proportionality factor p, satisfy 
an equation of Laplace of the form (18). The converse is also true. So we 
may state a theorem as follows : 

A necessary and sufficient condition that a point, whose pentaspherical co- 
ordinates are functions of two variables u, v, may generate a surface referred to 
its lines of curvature is that the pentaspherical coordinates satisfy an equation 
of Laplace. 

46. The correspondence between points of Ma and points on a hyper- 
quadric in S*. The correspondence with which this section is concerned is 
defined as follows. Let the pentaspherical coordinates x of a general point in 
ordinary metric space M 3 be interpreted as the protective homogeneous coordi- 
nates of a point in a linear space S*; then to the points of space MS correspond 
the points on the hyperquadric Q 3 represented by equation (22) in the space $ 4 , 
corresponding points having the same or proportional coordinates. Some of the 
essential features of this correspondence will be set forth in the present 
section, so that the correspondence may be applied in the next section to the 
theory of the lines of curvature. An account in English of this transforma- 
tion may be found in the treatise f by Coolidge on the circle and the sphere. 

To a sphere with the equation Sax = in space M 3 corresponds the inter- 
section of the hyperquadric Q 3 by the hyperplane represented in space $ 4 
by the same equation. The point P a which is the pole of this hyperplane 
with respect to Q 3 is called the second image of the sphere. The sphere is a 
point-sphere in case the point P a is on the hyperquadric Q 3 . Consequently 
we have the theorem: 

* Ibid., p. 221; or 2d ed. (1914), p. 273. 
t Coolidge, 1916. 2, p. 474. 



228 PROJECTIVE DIFFERENTIAL GEOMETRY 

To the spheres in ordinary metric space MZ correspond as second images the 
points in space 84, point-spheres in M s corresponding to points on the hyper- 
quadric Qs in St. 

Two spheres with the equations 2ax = 0, S6z = in space M s determine a 
pencil of spheres, the equation of a general sphere of the pencil being of the 
form 

(X, /z scalars) . 



This pencil of spheres corresponds to the intersection of the hyperquadric 
Q 3 by the plane represented in space /S 4 by the first two equations, and has 
for second image the polar line ab of the plane with respect to Q 3 . Hence 
we state the theorem : 

To the pencils of spheres in space M 3 correspond as second images the straight 
lines in space 84. 

A pencil of spheres in space M 3 belongs to one or the other of three classes 
according as the corresponding line ab intersects the hyperquadric Q 3 in 
one or the other of three possible ways. If the line ab intersects Q 3 in two 
distinct points, the corresponding pencil in space M 3 contains two point- 
spheres. The spheres of such a pencil cut their line of centers in pairs of 
points in an involution, of which the two double points are the two point- 
spheres. The point-spheres separate every pair of corresponding points of 
the involution harmonically, and consequently the point-spheres are not only 
conjugate points but are actually inverse points with respect to every sphere of the 
pencil. If, secondly, the line ab is tangent to the hyperquadric Q 3 , so that the 
condition (26) is satisfied, there is only one point-sphere in the pencil, all 
of the spheres of the pencil being tangent to each other at this point. Final- 
ly, if the line ab is a generator of the hyperquadric Q 3 , every sphere of the 
corresponding pencil is a point-sphere (see Ex. 1). 

It follows from the preceding paragraph that when a line in space Si 
intersects the hyperquadric Q 3 in two points P a , Pb, as indicated in Figure 
35, any point P c on the line a6, defined by 

(27) c = Xa-f M& (X, M scalars) , 

is the second image of a sphere in the pencil containing the two point- 
spheres corresponding to the points P a , Pb- These point-spheres are in- 
verse points with respect to the sphere c. Conversely, it can be shown that 
if two points a, b in space Af 3 are inverse to each other with respect to a sphere c, 
these two points correspond to two points on the hyperquadric Q 3 which are 
collinear with the point P c which is the second image of the sphere; consequently 
a relation of the form (27) is satisfied. In fact, the two points a, 6 in space 



46] 



METRIC AND AFFINE APPLICATIONS 



229 



M 3 can be regarded as point-spheres determining a pencil of spheres to 
which the given sphere belongs. 

A circle in space M 8 can be regarded as the intersection of two spheres 
Saz = 0, Sto = 0. Therefore to a circle corresponds the conic of intersection 
of the hyperquadric Q 8 by a plane. The polar line ab of this plane with 
respect to Q 8 can be regarded as the second image either of the circle or of 
the pencil of spheres intersecting in the circle. 

Pairs of circles in space M 8 can be classified according to the relations of 
two straight lines to each other and to the hyperquadric Q 8 in space 4. We 





FIG. 35 

shall not make an exhaustive classification, but shall confine ourselves to a 
few remarks. // two circles in space M$ correspond to two intersecting lines 
in space $4, the point of intersection of these lines is the second image of a 
sphere containing both circles. The plane of the two lines intersects the hyper- 
quadric Q 3 in a conic. The nature of this conic, and the relations of the two 
lines to it, give information about two circles on the same sphere. If two 
circles correspond to two lines that do not intersect, the lines determine a 
hyperplane 3 which cuts the hyperquadric Q 3 in a quadric surface. The 
nature of this quadric surface, and the relations of the two lines to it, give 
information about two circles not on the same sphere. Equation (25) shows 
that two spheres in space M$ are orthogonal in case their second images are 
conjugate points with respect to the hyperquadric Q 8 . Therefore the pole of the 
hyperplane /S 8 with respect to Q 8 is the image of a sphere orthogonal to every 
sphere through either of the circles, and hence orthogonal to both circles. 
So we have the theorem: 



230 



PROJECTIVE DIFFERENTIAL GEOMETRY 



In ordinary metric space there is just one sphere orthogonal to two circles 
that are not on a sphere. 

When a curve in ordinary metric space M 3 is defined by its parametric 
equations in pentaspherical coordinates, a question arises as to the geo- 
metrical significance of the derivatives of the coordinates of a point on the 
curve. This geometrical significance is made clear by the following remarks. 
Let us consider in space M 3 a point P x on a curve C whose parametric vector 
equation in pentaspherical coordinates is x = x(t). The derivatives x' of the 





FIG. 36 



coordinates x, calculated for the same value of t as the coordinates of the point 
P x , are the coordinates of a sphere which has its center on the tangent at the 
point P x of the curve C and which passes through P x orthogonal to C. In order 
to prove these statements regarding the sphere x r , let us refer to Figure 
36 and consider the curve Ci on the hyperquadric Q 3 in space $ 4 that corre- 
sponds to the curve C, and on C\ the point x that corresponds to P x on C. 
At the point x the tangent of the curve Ci is determined by the point x and 
the point x'. The point x', being in space $ 4 and collinear with two consecu- 
tive points of the hyperquadric Q 3 , is the second image of a sphere in space 
M 3 with respect to which two consecutive points of the curve C are inverse 
to each other. These two consecutive points determine the tangent of C 
at P x , which therefore passes through the center of the sphere. Conse- 



47] METRIC AND AFFINE APPLICATIONS 231 

quently the sphere passes through the point of contact P s of the tangent 
and is orthogonal to it, and hence is orthogonal to the curve C at the point 
P x . This completes the proof. 

The equation Sza/ = derived from equation (22) shows analytically a 
fact just demonstrated above, namely, that the sphere x' passes through 
the point P x . This sphere is a point-sphere in case So/ 2 = 0. Then the tan- 
gent of the curve C\ lies entirely on the hyperquadric Qa. In this case the 
tangent of the curve C is an isotropic line, and C is a minimal curve. 

The next problem is to determine analytically the osculating circle and the 
osculating sphere at a point of a curve. Let us consider again a curve C 
whose parametric vector equation in pentaspherical coordinates is x = x(t), 
and consider a point P x on C. Any sphere Sax = 0, whose coordinates a are 
functions of the parameter t, and which passes through the point P x , con- 
tains two consecutive points of C at P x if, and only if, 2az' = 0. Such a 
sphere is tangent to C at P x , and is orthogonal to the derivative sphere at 
this point. A two-point sphere a contains three consecutive points of C at 
P x in case also 2az" = 0. Such a sphere contains the osculating circle of C 
at P x . There is a pencil of such spheres, every one of which has coordi- 
nates a satisfying the three conditions just written. Finally, a three-point 
sphere a contains four consecutive points of C at P x in case also Sax'" = 0. 
The sphere a is then the osculating sphere at the point P x of the curve C, 
and its coordinates are found by solving the equations 

(28) Zaz = 0, 2az' = 0, Zaz" = , Saz'" = 

for the ratios of the coordinates a. The result for a general one of these 
coordinates can be written by means of a determinant of the fourth order, 

(29) a = (x,x',x",x'"). 

47. The lines of curvature. The correspondence studied in the last sec- 
tion will now be used to transform the theory of the lines of curvature on a 
surface S in ordinary metric space M 3 into the theory of a quadratic net N x 
in space >S 4 . There are two spheres, called principal spheres, associated with 
each point P x of the surface S. These spheres correspond to points in 
space 84 that generate two nets in the relation of a transformation of 
Laplace. The relation of the sequence of Laplace thus determined to the 
sequence of Laplace determined by the quadratic net N x that corresponds 
to the lines of curvature on the surface S will be examined. Then the inter- 
pretation in space M 3 of a transformation of Ribaucour applied to the 
quadratic net N x will be explained. 



232 PROJECTIVE DIFFERENTIAL GEOMETRY 

Let us consider in ordinary metric space M 3 a surface S which is not a 
sphere, plane, or isotropic developable, and whose parametric vector 
equation in pentaspherical coordinates is x=x(u, v), the lines of curvature 
being parametric. Since the coordinates x satisfy a quadratic equation of 
the form (22), and also satisfy an equation (18) of Laplace according to the 
last theorem of Section 45, it follows that the lines of curvature on the surface 
S in space M 3 correspond to a quadratic net N x on the hyperquadric Q 8 in 
space S*. 

We shall now show that at each point P x of the surface S there exist two 
spheres each of which has the characteristic properties that it is tangent to S at 
P x , contains the osculating circle of a line of curvature at P x , and, as P x varies 
along that line of curvature, has a characteristic circle which is tangent to S at 
P x . These spheres will be called the principal spheres of S at P x . To make 
the demonstration* let us consider any curve C through the point P x on the 
surface S. There is a one-parameter family of spheres tangent to S at P x , 
and among them there is just one sphere that contains the osculating circle 
of C at P x . The coordinates z of this sphere satisfy the conditions 

(30) 2zz = , 2zx u = , 2zx v = , 2zz" = . 

As the point P x varies along the curve (7, the characteristic circle of the 
sphere z is the intersection of this sphere and the sphere z' . Conditions 
necessary and sufficient that the sphere z', and hence the characteristic 
circle, be tangent to the surface $ at the point P x are 

(31) Zz'z = , 2z'z u = , 2z'x v = . 

The first of these conditions can be obtained by differentiating the first, 
and using the second and third, of equations (30). In the presence of the 
equations obtained by differentiating the second and third of (30), the 
second and third of the conditions (31) are equivalent to S2(z w )' = 0, 
SzOct,y = 0. The last of equations (30) is a consequence of these two new 
conditions and the second and third of (30). Thus the following independ- 
ent conditions have been imposed on the sphere z and the curve C: 

(32) 2zx = , 2zx u = , 2zx v = , Sz(z w )' = , 2z(x v )' = . 
Elimination of z therefrom gives a condition on the curve C, 

(33) (x, x, x v , x uu du+x uv dv, x uv du+x vv dv) =0 . 
* Tzitz&ca, 1924. 3, pp. 217-18. 



47] METRIC AND AFFINE APPLICATIONS 233 

No use has yet been made of the fact that the lines of curvature are para- 
metric. By means of the Laplace equation (18), equation (33) can be re- 
duced to 

(34) (x, #, x v , x uu , x vv )dudv = Q . 

Since the surface S is not a sphere or a plane, the coordinates x do not satisfy 
a linear algebraic equation with constant coefficients. Moreover, since S 
is not an isotropic developable, the coordinates x do not satisfy a second- 
order linear differential equation (see Ex. 4) that is independent of the 
Laplace equation (18). Therefore the first factor of equation (34) does not 
vanish, and the curve C must be one of the two parametric lines of curvature 
through the point P x . Placing v = const, in equations (32) we may easily 
show that the resulting equations determine a unique sphere z. Similarly, 
placing u = const, in (32) we may show that the equations obtained deter- 
mine a unique sphere z-\. In fact, the two systems of equations in question 
are the following: 



( Szz = 0, 
(oo) ^ 

1 2z-ix = , 



, 2z-ix u = , 2z-ix vv = . 



Actual solution of the first system for the ratios of the coordinates z, and of 
the second system for z-i, completes the existence proof and yields the fol- 
lowing theorem. 

The coordinates z of the principal sphere corresponding to the u-curve, and 
the coordinates z_i of the principal sphere corresponding to the v-curve at a 
point P x of a surface S, referred to its lines of curvature, are given by the 
formulas 

(36) Z = (X, X u , X v , X uu ) , 2-1 = (X, X v , Xu, X vv ) . 

Making use of the familiar notation for Laplace transforms, we may 
write 

Z = (Z_ 2 , X-i, X, Xi) , Z_i= (X-i, X, Xi, Z 2 ) 

An application of the theory of polar sequences of Laplace developed in 
Section 33 leads to the following conclusions (see Fig. 37). As u, v vary, the 
point z, which is the second image in space S* of the principal sphere z, generates 
a net N z which is one net of the polar sequence, with respect to the hyperquadric 
Qs, of the sequence determined by the net N x . The point z-\ generates the minus- 
first Laplace transform of the net N z , as the notation indicates. 



234 PROJECTIVE DIFFERENTIAL GEOMETRY 



We shall now prove that the congruence of lines zz_i is conjugate to the 
net N x . For the purpose of the demonstration* let us suppose that the pro- 





FIG. 37 

portionality factor of z has been chosen so that Sz 2 = 1. Differentiating this 
equation and the first three of (35) with respect to u and simplifying, we 
find 

Sz tt z = , 2z u x u = , 2z u x v = , Sz u z = , 



so that z u = (x, x u , x v , z); we also have 



so that x = (x, x u , x v , z). Consequently we obtain 

(37) z tt = ^, 

where ^ is a proportionality factor. From the equations 

2z t x = , 2z v x u = , 2z v x uu = , 2z v z = , 
* Tzitz6ica, 1924. 3, pp. 218-19. 



47] METRIC AND AFFINE APPLICATIONS 235 

and from the two groups of equations 

?,x v x = , 2x v x u = , 2x v x uu +a2xl = , 2x v z = ; 



which imply 



it follows that 

(38) z, = ^i, 

where <p is a proportionality factor. From equations (37), (38) it follows 
that the function <p is a solution of the adjoint of the Laplace equation (18) , 
and the proof is finished by applying the theory developed at the close of 
Section 32 for determining all congruences conjugate to a given net. 

Let us consider the net N x on the hyperquadric Q 3 in space 4 that corre- 
sponds to the (parametric) lines of curvature on a surface S x in space M 3 . 
Let us apply to the net N x a transformation of Ribaucour to obtain a net 
N y , also on the hyperquadric Q 3 of course, and corresponding to the lines 
of curvature on a surface S y in space M 3 . It is clear that the lines of curvature 
on the surfaces S x and S v correspond, and we are going to prove* that S x , S v 
are the two sheets of the envelope of a two-parameter family of spheres, a general 
sphere of the family touching the two surfaces in corresponding points P XJ P v . 
For this purpose, referring to Figure 38, let us observe that, since the nets 
N x , Ny are conjugate to one congruence, these nets are harmonic to another. 
Therefore, if the point of intersection of the lines xx u , yy u is denoted by p and 
that of the lines xx v , yy v by a, the points p, a are the focal points of a gener- 
ator of the harmonic congruence. Using the notation of equation (V, 55), 
we find 



(39) (P = Xx,-X u x=- 

I a = Xx v X v x = 



= Xx u -X u x = -(Xy u -X u y)/Y , 
X v x=-(Xy v -X v y)/Y . 



The points p, a are the second images of two spheres in space M 3 , both of 
which are orthogonal to S x at P x and to S v at P v . For, the easily verified 
conditions 

(40) 

* Ibid., p. 220. 



236 



PROJECTIVE DIFFERENTIAL GEOMETRY 



show that both spheres pass through P x and through P y ; moreover, the 
sphere p is orthogonal to the w-curve at the point P x , and is also orthogonal 





FIG. 38 

to the w-curve at the point P v , while the sphere <r is orthogonal to the v- 
curves at these points. The spheres p, a intersect in a circle orthogonal to 
the surface S x at P x and to the surface S v at P v . The sphere r defined by 



(41) 



Sn/ = 



passes through the point P x orthogonal to this circle, and likewise through 
the point P v orthogonal to the same circle. Therefore the sphere r is tangent 
to the surface S x at the paint P x and to the surface S v at the point P V) so that, as 
u, v vary, the sphere r envelops the surfaces S x , S v . This completes the proof. 
We conclude with the definition of isothermic surfaces, and a theorem 
concerning them. The parametric curves on a surface in space 7kf 3 have 
been seen to be an orthogonal net in case F = 0. The parametric curves on a 
surface are said to form an isothermally orthogonal net in case 



(42) 



48] METRIC AND AFFINE APPLICATIONS 237 

If the lines of curvature on a surface form an isothermally orthogonal net, 
the surface is by definition isothermic. Then equation (16) has equal La- 
place-Darboux invariants. Consequently we may state the following con- 
clusion: 

A surface in ordinary metric space M s is isothermic if, and only if, its lines 
of curvature correspond to a conjugate net with equal invariants on the hyper- 
quadric Q& in space 84. 

48. A local trihedron at a point of a surface in M&. For the purpose of 
investigating certain aspects of the metric differential geometry of a surface 
in ordinary metric space M 8 it is convenient to introduce a local trihedron 
of reference at a general point of the surface. In this section we use the tri- 
hedron whose edges are the tangents of the lines of curvature and the 
normal at the point of the surface. This trihedron is in some respects analo- 
gous to the local tetrahedron employed in Chapter III, and seems especially 
suited for studying metric properties of certain protectively defined con- 
figurations associated with a point of a surface. For example, metric theo- 
rems are obtained concerning the quadrics of Darboux, union curves, and 
the axis and ray congruences of a conjugate net. 

Let us consider a non-developable surface S, which is not a sphere, and 
which is defined analytically as in Section 44; let us suppose that the lines 
of curvature on S are parametric, so that F=D' = 0, DD'VO. Some of the 
formulas of that section simplify considerably under this assumption. For 
instance, equations (15) give 



(43) R 

The last two of equations (11) become 

(44) X u = -Xu/Ri , X v = -x,/Ri , 

while the second of equations (11) becomes equation (16), since the Christof- 
fel symbols (10) are now expressed by the simpler formulas 



,1} =(log E) , 2{22, 2} =(log G). , 
2{21, 1} -(log E) v , 2(12, 2} = (log G). , 
2(22, 1} = -Gu/E , 2(11, 2} = -ft/G . 

The equations (13) of Codazzi can now be written in the form 
( 



238 



PROJECTIVE DIFFERENTIAL GEOMETRY 



As a local trihedron of reference (see Fig. 39) at a point (x, y, z} of the sur- 
face S referred to its lines of curvature let us take the origin at this point, 
the -axis along the u-tangent, the rj-axis along the v-tangent, and the f-axis 






FIG. 39 



along the surface normal through the point. If x, y, z are the general co- 
ordinates of a point having local coordinates , r;, f , the equations of trans- 
formation between the general and the local trihedrons can be written in 
the form 



(46) 



The equations of the inverse transformation are 

t = (x-x,x v /GW, X 
l = (x-x, X , 



in which parentheses indicate determinants of which only the first row is 
written in each case, the second and third rows being obtained from the 
first by replacing x, Xbyy, Y and then by z, Z. 

The local equations of the tangent line at a point P of a curve C defined by 
v = v(u) on a surface S are 



(47) 



48] METRIC AND AFFINE APPLICATIONS 239 

where 

(48) tan 6 = t/(G/#) 1 / 2 (v f = dv/du) , 

and 6 is the angle that the tangent of the curve C makes with the -axis. In 
particular, the equations of the asymptotic tangents at the point P are 



(49) f = 

The local equation of the osculating plane at the point P of the curve C is 

(50) (D+D" 
where p is defined by placing 



}-{22, 2}X 2 -{22, !>'] . 

Then 1/p is what is known as the geodesic curvature of the curve C at the 
point P. The geodesic curvature vanishes at every point of the curves called 
geodesic curves. In particular, for the geodesic curvatures I/pi, 1/P2 of the 
parametric lines of curvature C u , C v at the point P we find* the expressions 

(52) 1/ P1 = - (log E) V /2GV* , 1/ P2 = (log 
Equation (50) can be written also in the form 

(53) E^D+D'^M tan e-i 



For the purpose of obtaining metric results concerning some of the pro- 
jectively defined configurations discussed in Chapter III, we shall employ 
power series expansions for the local coordinates , 77, of a point near a point 
P(0, 0, 0) on a surface S, in terms of the increments Aw, At; corresponding to 
displacement on S from the point (0, 0, 0) to the point (, 77, f). The calcu- 
lation of these expansions is so similar to the calculation of the series (III, 
15) that it will not be reproduced here. The results are as follows: 



0+ {22, l)At; 2 )/2+ 
, l,2}A W AH-{22,2)At, 2 )/2+. 

( ) i^ /T^..Q , /A..OW , r/r* , n^ 1}D)AW 3 + 3{12, 



11 Eisenhart, 1909. 1, p. 134. 



240 PROJECTIVE DIFFERENTIAL GEOMETRY 

The equation of any quadric having contact of the second order with the 
surface S at the point P is easily shown to be of the form 

(55) ?/Ri+if/R*+2f(-l+k+kn+ktf=0 , 

where & 2 , &s, & 4 are arbitrary. This quadric is not a parabola if the function 
/ defined by 



(56) /= (2fc 4 - Rik\ - 

is not zero; then the center of the quadric is the point 

(57) (-*//*, -*i//Bi, 



The center is on the normal of the surface at the point P in case #2 = & 3 = 0. 
The equation of the curves of Darboux on a surface is found to be 



(58) D&'du* - WWdtfdo - ZD"&duM*+D"WW = , 
where the functions 93', (' are given by the formulas 

(59) 8' = (log R*/R Z ) V , 8<' = (log flj/ft). . 

The quadric (55) is a quadric of Darboux in case 

(60) 4^/2^ = (i g K ) u t 4i/ 2/b3 = (log K) v , 



where K is the total curvature defined by the first of equations (15). Hence 
we deduce the theorem*: 

The centers of the quadrics of Darboux at each point P of a surface S are on 
the normal of Sat P if, and only if, the surface S has constant total curvature. 

Any line l\ through a point P(0, 0, 0) and not in the tangent plane, f = 0, 
can be regarded as joining P to a point ( a, b, 1), where a, b are func- 
tions of u, v. By means of equation (50) the union curves of the congruence 
TI generated by h are found to have the differential equation 



(61) {' 



, 1}-2{12,2}X 
-{22, 2}+2{21, l})t 



If the line li is normal to the surface then a= 6 = and the union curves are 
the geodesies on the surface. Thus we have proved the theorem: 

* Fubini and Cecil, 1926. 1, pp. 177-78. 



49] METRIC AND AFFINE APPLICATIONS 241 

The union curves of the normal congruence of a surface are the geodesies on 
the surface. 

The equations of the osculating planes of the lines of curvature (7 tt , C v 
at a point P are respectively 

(62) pirj-flif =0 , p 2 +# 2 f = 0, 

where Ri, Ri are given by equations (43), and pi, p 2 by (52). Therefore the 
axis of a point P(0, 0, 0) on a surface, with respect to the lines of curvature, 
joins P to the point (a, 6, 1) for which 

(63) a = # 2 /p2, 6=-#i/Pi. 

Moreover, the ray of the point P with respect to the lines of curvature intersects 
the tangents of the lines of curvature C u , C v through P in the points whose 
general coordinates are given by the respective expressions 

(64) x-p&u/EV 2 , x + plXv /(}i/2 9 

as can be verified by showing that the ^-derivative of the first expression is 
proportional to x u , and that the ^-derivative of the second is proportional 
to x v . Additional results related to this section may be found in Exercises 
9, . . . , 13, 16, 21, 22, 24. 

49. The transformation between cartesian coordinates and Fubini's 
normal coordinates. Besides the method used in the last section for study- 
ing metric properties of protectively defined configurations in the geometry 
of surfaces in ordinary space, there is still another method, which will be 
employed in this section. This method rests on the following observation. 
If x, y, z are the ordinary cartesian coordinates of a point P on a surface, 
then the homogeneous cartesian coordinates x, y } z, 1 can be regarded as a 
special kind of protective homogeneous coordinates of P, and there exists 
a function X such that Xz, Xy, Xz, X are Fubini's normal coordinates of P, 
which satisfy equations (III, 6) when the asymptotic curves on the surface 
are parametric. After simplifying the formulas of Section 44 by choosing 
the asymptotic curves as parametric, we shall actually find a formula for 
the factor X. The result will be applied in a brief discussion of the projective 
normal and the first directrix of Wilczynski. 

Let us consider a non-ruled surface S defined as in Section 44, and let us 
suppose that the asymptotic curves on S are parametric, so that D = D" = 0, 
D'j^O. Some of the formulas of that section simplify considerably under 
this assumption. If a function fc 2 is defined by placing 

(65) H 2 



.242 PROJECTIVE DIFFERENTIAL GEOMETRY 

then for the total curvature K we have the expressions 

(66) X=-Z>'Y# 2 =-l//c 2 . 
The equations (13) of Codazzi become 

(67) (log >')= {11, 1}-{12,2} , Oog0').H22,2}-{21, 1} , 
and it is easy, by means of the formulas (10), to verify the identities 

(68) (log#) tt ={ll, 1} + {12,2} , (logff).H22,2} + {21, 1} . 
Consequently we obtain 

(69) (log fc) tt = 2(12,2} , (log/c) v = 2{21, 1} . 

We proceed to discover the formula for the multiplier X that converts 
cartesian coordinates into Fubini's normal coordinates. The effect of a 
transformation x = \x on the coefficients of equations (III, 6) is given by 
equations (III, 4). Let the coordinate system be specialized by choosing 
for the tetrahedron of reference the tetrahedron whose faces are the three 
cartesian coordinate planes and the plane at infinity, and by selecting the 
unit point appropriately, so that the coordinate system is cartesian. Fur- 
ther, let the coordinates x be the cartesian coordinates x, y, z, 1. Since X 
must now be a solution of equations (III, 6), it follows from the first two 
of equations (III, 4) that p q = Q. Moreover, the first and third of equa- 
tions (11) and the last three of equations (III, 4) now give 



f {11, 1} = tt = M -2(log X) , {11, 2} =/3 = /3 , 
I {22, 1) =7 = 7, (22, 2}=0; = t ,- 

Equations (68), (69), (70) imply 

(log #) u = tt -2(log X).+dog *)/2 , 
(log ff). = 0,-2(log X).+(log fc)./2 . 



These equations determine the function X, except for a constant factor; 
integrating them we obtain the desired formula, which can be written in two 
equivalent forms, 



(71) X=^/V' 1/2 fc 1/4 = ({ll, 2} (22, 



49] METRIC AND AFFINE APPLICATIONS 243 

We have thus established the result: 

Multiplication of the cartesian coordinates x, y, z, 1 by the factor X given in 
(71) converts them into Fubini's normal coordinates. 

As an application of this result, we shall show how it can be used to study 
metric properties of the projective normal. When the coordinates are Fubini's 
normal coordinates, the projective normal at a point x of a surface joins 
this point to the point x uv . After a transformation x = \x, the projective 
normal joins the point \x to the point (\x) uv . If X is given by (71), and if the 
coordinates x are the cartesian coordinates x, y, z, 1, it follows that the pro- 
jective normal joins the point (x, y, z, 1) to the point [(Xz) ut> , (\y) uv , (Xz) U v, 
\ uv \. The point at infinity on the projective normal has the homogeneous car- 
tesian coordinates 



, 0] . 

The direction cosines of the projective normal are proportional to the first 
three of these coordinates, and hence are proportional to the expression 

(log Xfc^zu+Oog W^+D'X 

and the two other expressions obtained by substituting y, Y and z, Z in 
place of x, X. The cartesian equations of the projective normal could 
easily be written, since its direction cosines and the coordinates of a point 
on it are known. 

A condition necessary and sufficient that the projective normal may coincide 
with the metric normal is found, by equating to zero the coefficients of x U} x v 
in the last expression above, to be 



(72) X/c!/ 2 = c (c = const.) . 

Comparing the value of X thus determined with the value given by (71) we 
arrive at a condition on a surface at every point of which the two normals 
coincide, namely, 



More generally, the direction cosines of any canonical line of the first 
kind are proportional to the expression 



and the two symmetrical expressions in y, Y and z, Z, in which k 1 is the 
constant k of Section 20 and <p, ^ are defined by equations (III, 9). This 



244 PROJECTIVE DIFFERENTIAL GEOMETRY 

line coincides with the metric normal in case the coefficients of x u and x v 
vanish. If the canonical line* is the directrix of Wilczynski, for which 
k' = 1/2, the conditions that this line may coincide with the metric normal 
can be written in the form 

(73) (log fc)* = 2(log $D') V , (log fc)u=2(log 7#')u . 

It follows that (log 0/7)ut> = 0. A surface for which this partial differential 
equation is satisfied is called isothermally asymptotic. After a transforma- 
tion of parameters we have 

/3 = y 9 & = cpPD' 2 (c = const.) . 

50. Affine geometry of surfaces. In this section we do not attempt to 
establish a theory of the invariants of surfaces under the general affine 
transformation in ordinary space, but consider only such affine transforma- 
tions as leave one point, the origin, invariant. A completely integrable 
system of differential equations is set up which defines a surface except for 
an affine transformation of this kind, and an interesting property of a pair 
of integral surfaces of this system of equations is established. A special 
class of surfaces, each of which has a property called the property of Tzitzeica, 
is defined, and the section closes with a formula connecting the coordinates 
used in the affine theory with Fubini's normal coordinates. 

Let us first of all prepare to exlude a certain special class of surfaces. If 
the cartesian coordinates z, y, z of a point on a surface S in ordinary space 
are given as functions of two variables u, v and satisfy a differential equa- 
tion of the form 



whose coefficients a, 6, c are scalar functions of u, v and are not all zero, then 
integration of this equation shows, as in Section 8, that the coordinates 
x, y, z can be expressed in the form 

(74) x = */i(0 , y = */t (fl , z = */,(0 , 

where <p, t are functions of u, v. Elimination of <p, t leads to a homogeneous 
algebraic equation in x, y, z. Therefore the surface S is a cone with its vertex 
at the origin. Such surfaces will be excluded from the following discussion, 
unless the contrary is explicitly stated. 

* Fubini, 1927. 8. 



50] METRIC AND AFFINE APPLICATIONS 245 

When the coordinates x, y, z of a general point on an unspecialized sur- 
face S are given as functions of two variables u, v it is possible to deter- 
mine* the coefficients of the equations 



x vv =qx+yx u +8x v 

so that each of x, y, z will satisfy each of the equations. For example, if 
each of x, y, z is substituted in turn in the first equation with the coefficients 
p, a, /3 regarded as unknown, the resulting system of three linear algebraic 
equations can be solved uniquely for these coefficients. The other coeffi- 
cients can be determined similarly. Then S is an integral surface of the 
system (75). This system is of the same form as the system (IV, 26), of 
which the integrability conditions are equations (IV, 27). 

A transformation whose equations in cartesian coordinates are linear and 
homogeneous with constant coefficients, of the form 



is an affine transformation leaving the origin invariant. Any surface ob- 
tained from an integral surface of system (75) by such a transformation is 
also an integral surface of the system; conversely, system (75) with its 
integrability conditions satisfied determines a surface except for such an 
affine transformation. 

Using the cosines X, F, Z defined by the formulas (6) and the second 
fundamental coefficients D, D', D" defined by (9) we find, for an integral 
surface S of system (75), 

(76) D = p2Xx, D' = c2Xx, D" = q?Xx . 

If 2Xx = 0, the surface S is a plane passing through the origin. Such surfaces 
being excluded, the differential equation (8) of the asymptotic curves on the 
surface S becomes 

(77) pdu?+2cdudv+qdv* = . 

If the asymptotic curves on the surface S (supposed non-developable) are 
parametric, then p = #=0, c?^0, and the first two of the integrability con- 
ditions (IV, 27) become 

(78) (log c) tt = a-6 , (log c), = 5-a . 
* Tzitz&ca, 1924. 3, p. 244. 



246 PROJECTIVE DIFFERENTIAL GEOMETRY 

Consequently we have (a b) v = (B d) u ; but the third and fourth of the 
conditions (IV, 27) imply (a+6) v = (5+a) u . One deduces immediately the 
equations 

(79) a v = , b v = a u , 

and hence a theorem : 

When the asymptotic curves on a non-developable integral surface of system 
(75) are parametric, the Laplace equation in this system has equal invariants. 
We shall now establish a metric property* of any pair of non-developable 
integral surfaces S, S' of system (75). Let d, d f denote the distances from 
the origin to the tangent planes at corresponding points of S, S', and let 
K, K' be the total curvatures of these surfaces at these points. We propose 
to prove that the surfaces S, S' have the property expressed by the equation 



(80) K/d*+K'/d f * = const. 

First of all, we observe that the distance d from the origin to the tangent 
plane (2) at a point (x, y } z) of the surface S is given by 



(81) d=2Xx = D'/c=(x, x u , x v )/H . 

If the surface S is referred to its asymptotic curves, the total curvature K 
of S at the point (x, y, z) is given by 



(82) K = - 
Consequently we find 

(83) K/d* = - c 2 /^ 2 = - c 2 /(z, x u , x v 
Let us define a function n by placing 

(84) 
so that 
(85) 

Logarithmic differentiation of equation (85) and use of equations (75), (78) 
lead to the conditions 



(86) (logn) u = 6, (logn),-a. 

* Ibid., p. 246. 



50] METRIC AND AFFINE APPLICATIONS 247 

Consequently log n is determined by system (75), except for an additive 
constant, and the function n is determined except for a multiplicative con- 
stant. Equation (84) now shows that the ratio K/d* is determined by sys- 
tem (75), except for a constant factor. The ratio K '/d'* is determined by 
system (75) in the same way, except for a constant factor. Therefore the 
statement made in equation (80) is true, as was to be proved. 

It follows from the preceding paragraph that if K/d^ const., then also 
K'/d'* = const. Therefore the property expressed by the equation K/d 4 = 
const, is of an affine nature. Equations (84), (86) show that all non-develop- 
able integral surfaces of system (75) have this property if, and only if, a = b = 0. 
We shall now discover an affine geometric characterization of such surfaces. 
The parametric equations of the ruled surface R u of w-tangents constructed 
at the points of a fixed v-curve of such a surface are 



z=z+tz u , 

wherein u = const, and the parameters are t, v. The equation 

((-x, x t , x v )=Q 
of the tangent plane at a point (x, y, z) of the surface R u reduces to 

( x, x u , x v )+t( x, x u , x uv )=Q . 

Consequently the equation of the tangent plane at a point at infinity (or an 
asymptotic tangent plane) of the surface R u is 

(87) ({-a, x ut z v )=0. 

This plane passes through the origin if, and only if, 6 = 0. Thus we know a 
geometrical meaning for the condition 6 = 0. Similarly, using the ruled sur- 
face R v , defined as R u but with u and v interchanged, we obtain a meaning 
for the condition a = 0. Therefore a surface has the property expressed by the 
equation K/d 4 = const, if, and only if, each ruled surface of the tangents of the 
asymptotic curves of one family, constructed at the points of a fixed asymptotic 
curve of the other family, cuts the plane at infinity in a curve such that the tan- 
gent planes of the ruled surface at the points of this curve pass through the origin. 
But these tangent planes envelop the asymptotic developable of the ruled 
surface. Moreover, we shall call the property expressed by the equation 
K/d* = const, the property* of Tzitzeica, because it has been studied so ex- 

* Ibid., p. 250. 



248 PROTECTIVE DIFFERENTIAL GEOMETRY 

tensively by him. We are enabled thus to conclude with the following 
proposition. 

A non-developable surface in ordinary space has the property of Tzitzeica if, 
and only if, the asymptotic developables of the ruled surfaces of tangents of the 
asymptotic curves of each family, circumscribing the surface along the asymp- 
totic curves of the other family, are cones all of which have a common vertex. 

The same method that was used in the preceding section can be used 
here to establish the transformation between the coordinates x, y, z of the 
present section and Fubini's normal coordinates. We take the asymptotic 
curves as parametric, and suppose that the surfaces considered are not ruled. 
If X is the factor by which the homogeneous coordinates x, y, z, 1 of the 
affine theory must be multiplied in order to convert them into Fubini's 
normal coordinates, we find 

(88) X 2 = he 9 /en (h = const. ; 6 = log 7) , 
where n satisfies the conditions (86). 

61. Surfaces with indeterminate directrix curves. The directrix curves 
on a surface S in ordinary projective space are defined to be the curves in 
which the developables of the first directrix congruence of S intersect S; 
the same curves also correspond to the developables of the second directrix 
congruence of the surface S. The purpose of this section is, first of all, to lay 
the foundations for a purely projective theory of those surfaces on each of 
which the directrix curves are indeterminate. On such a surface every 
curve is a directrix curve, and therefore corresponds to a developable of each 
directrix congruence. Then by means of the transformation introduced at 
the close of the last section, which permits to pass from the projective 
theory to the special affine theory of surfaces, it will be proved that every 
surface, not of a certain special type, whose directrix curves are indetermi- 
nate can be projected into a surface having the property of Tzitz&ca. 

The differential equation of the directrix curves on an integral surface S 
of system (III, 6) is equation (III, 39) when the functions a, b therein have 
the values shown in equations (III, 48). These curves are indeterminate 
in case the three coefficients in their equation vanish. From the vanishing 
of the second coefficient we obtain (log j3/7) ww = 0, so that by a change of 
parameters we have 18 = 7. This means that the surface S is isothermally 
asymptotic. The vanishing of the other two coefficients gives two conditions 
which, when =7, reduce to 

(89) p=/uu 



51] METRIC AND AFFINE APPLICATIONS 249 

where for abbreviation we have defined/ by placing 

(90) 2/=31og/3. 

Substitution of these expressions for the coefficients p, q in the first two of 
the integrability conditions (III, 10) gives two partial differential equations 
for the function/, each of which can be integrated once. The result of this 
integration is that must satisfy the equation 

(91) (log 0) uv = p+k/p (k = const.) . 

The third integrability condition is satisfied identically. Thus we are led 
to the following conclusion. 

// 18 is a solution of equation (91), if p, q are defined by equations (89), and 
ify = P, then all of the integrability conditions (III, 10) are satisfied, and every 
integral surface of system (III, 6) is a surface* with indeterminate directrix 
curves. 

Equation (III, 40) shows that, when the directrix curves are indetermi- 
nate, the two foci of the directrix d\ of the first kind at a point x coincide in 
the point z given by 

(92) z = (f u f v -Zp 2 /2-k/2(3)x-f v x u -f u x v +x uv . 

That the point z is fixed is evident geometrically, and is demonstrated ana- 
lytically by the formulas 

(93) 3z u =-/*, 3s.= -/rf, 

which follow from differentiating z and reducing the results. All the direc- 
trices of the first kind pass through the fixed point z. Dually, all the directrices 
of the second kind lie in a fixed plane. This plane is determined by the three 
points 

Xu JU% 1 %V JV% ) %UV Jw% JU%V 

Therefore the equation of this plane referred to the local tetrahedron 
x, x u , x v , x uv , with suitably chosen unit point, is 



(94) 

The point y in which the line xz meets the fixed plane (94) is given by 
(95) y=z-kx/0. 

* WUczynski, 1914. 3. 



250 PROJECTIVE DIFFERENTIAL GEOMETRY 

It is evident now that the fixed point and plane are in united position if, and 
only if, /b = 0. In this case equation (91) shows that the asymptotic curves 
on the surface S belong to linear complexes. In what follows we shall sup- 
pose that k 7**0. 

We shall next show that a surface S with indeterminate directrix curves 
has the property that each ruled surface of the tangents of the asymptotic curves 
of one family, constructed at the points of a fixed asymptotic curve of the other 
family, intersects the fixed plane (94) in a curve such that the tangent planes of 
the ruled surface at points of this curve envelop a cone with its vertex at the 
fixed point z. For this purpose it is useful to observe that the ^-tangent at 
a point x of the surface S meets the fixed plane (94) at the point x u f u x. 
The tangent plane at this point of the ruled surface generated by the u- 
tangent as v varies is determined by this point, the point x, and the point 
(x u f u x)v, or else by the points x, x u , x uv f u x v . To complete the proof it is 
sufficient to remark that the point z given by equation (92) is linearly de- 
pendent on these points. 

The theory of surfaces with indeterminate directrix curves, as outlined 
in the foregoing paragraphs of this section, has been purely projective. 
But we are now going to employ the transformation mentioned at the end 
of the preceding section to pass to the affine geometry of these surfaces. In 
particular, we shall show that every surface with indeterminate directrix 
curves (k^Q) can be projected into a surface having the property of Tzitzeica, 
by choosing the fixed point as the origin and projecting the fixed plane into 
the plane at infinity. 

After a transformation x = \x, equation (92) becomes 

X) v -/ v (log X) u +X wv /X]z 

(log X),]x tt -[/ tt -(log \) u ]x v +x uv . 

The points x u fuX, x v f v x, which are on the directrix d z of the second kind 
at the point x, are represented by the expressions 

(97) x u - [/u- Gog \) u ]x , x v -[f v - (log \) v ]x 

after the transformation. If the function X is defined by equation (88), if 
the coordinate system is specialized so that it is cartesian, if the coordinates 
x are homogeneous cartesian coordinates x, y,z, 1, and if the fixed point z of 
equation (96) is chosen as the origin, then the non-homogeneous cartesian 
coordinates x, y, z must satisfy the equation that results when the right 
member of equation (96) is set equal to zero. Finally, if the fixed plane con- 
taining the directrices of the second kind is projected into the plane at 



51] METRIC AND AFFINE APPLICATIONS 251 

infinity, the two expressions (97) must vanish when 1 is substituted therein 
in place of x; consequently we have 

(98) (logX) tt =/u, (logX).-/.. 

The effect of the transformation under consideration on equations (III, 6) 
is easily calculated. Thus we find that the coordinates x, y, z satisfy a sys- 
tem of equations* of the form (75), namely, 



x vv =(3x u -(log0) v x v 

wherein /3 is a solution of equation (91) with fc^O. Since the terms in z tt , x v 
are missing from the second of equations (99), it is seen that a = 6 = in 
the notation of system (75). Therefore all integral surfaces of system (99) 
have the property of Tzitzeica. Thus we have completed the proof promised 
in the last paragraph. 

EXERCISES 

1. If every sphere of a pencil in ordinary metric space Ms is a point-sphere, the 
line of centers is an isotropic (minimal straight) line. Therefore the isotropic lines 
of space Ms correspond to the rectilinear generators of the hyperquadric Qa in 
space 84. 

2. The circle common to all the spheres of a pencil of the first or second class in 
space M 3 lies in a plane (called the radical plane of the pencil) . If this plane is taken 
as the plane x = 0, the equation of a general sphere of the pencil is 

X 2 _|_ y* _|_ 2 2 __ 2g X -j- c = (c = const . ; g = parameter) . 

The circle is real, a point-circle, or imaginary according as the pencil contains two 
imaginary, one real, or two real point-spheres. Prove analytically that in the last 
case the polar plane of each point-sphere with respect to any sphere of the pencil 
passes through the other point-sphere. 

3. Two spheres are orthogonal in case one of them contains two points that are 
inverse points with respect to the other sphere. 

4. If the pentaspherical coordinates of a point on a surface S referred to its 
lines of curvature satisfy an equation of Laplace and an independent differential 
equation of the second order, the corresponding net N x on the hyperquadric Q 3 lies 
on a developable surface, and S is an isotropic developable; that is, S consists of 
isotropic lines tangent to a minimal curve. 

* Fubini and Cech, 1926. 1, p. 167. 



252 PROJECTIVE DIFFERENTIAL GEOMETRY 

5. The sphere r defined by equations (41) has for second image in space $ 4 a 
point which, as u, v vary, generates a net which has a Laplace equation of the form 
ruv = Ar u +Br v when the proportionality factor of r is chosen so that Sr*= 1. 

TziTziiCA, 1924. 3, p. 221 

6. A transformation by inversion from a surface S x to a surface S v with respect to 
a sphere z in space M* corresponds to a transformation of Ribaucour from a net N x 
to a net N v on Qa in $4, the conjugate congruence of this transformation being a 
bundle of lines with center at the point z. Hence lines of curvature are preserved 
under a transformation by inversion. 

7. An inversion with respect to a sphere in space M 3 carries an isothermic surface 
into an isothermic surface. 

8. When the lines of curvature are parametric on a surface S x in space Ms, if 
the v-curves are spherical, the Laplace sequence determined by the corresponding 
net NX in space S* terminates in the ^-direction according to the case of Goursat. 

TZITZEICA, 1924. 3, p. 239 

9. The differentials of the direction cosines of the three edges of the local tri- 
hedron composed of the tangents of the lines of curvature and the normal at a 
point of a surface are given by the formulas 



, 2}dv)x.+DX 
, l}du+ (22, l}dv)xu+D"X dv]/G 1 / 2 , 



10. Use the formula 

x = x+r( - axu/E 1 /* - bx v /GV*+X) 



for any point x on a line h to show that the differential equation of the IYcurves of 
the congruence of lines h is 



and that the foci of the line li are determined by solving 

(Pi*Si~riQi)r2-(5 1 +P 1 )( 
where Pi, Qi, 2\, Si are defined by the formulas 



!, 2}a+Dab , 
2, 2}a 

Hence show that the Pi-curves of the normal congruence are the lines of curvature. 



METRIC AND AFFINE APPLICATIONS 253 

11. The reciprocal line l z of the line h of Exercise 10, with respect to any quadric 
of Darboux, crosses the parametric tangents of the lines of curvature at the points 
p, <r defined by 



where 

1/X= -a/ 

12. Use the formula 



for any point x on a line lz (see Ex. 11) to show that the differential equation of the 
recurves of the congruence of lines I* is 



and that the foci of the line Z 2 are determined'by solving 

Q 2 D*GW+ (St-P 2 )DD"(EG) l /*\nr(l -r) - TJ)'"*Ef(l -r)'=0, 
where Pz, Qz, T 2 , S 2 are defined by the formulas 

(log X 
(log \ 
(log M) 



For the ray of the lines of curvature X= P2, /*= PI; the ray curves of the lines of 
curvature on a surface form a conjugate net if, and only if, the surface is isothermic. 

13. The formulas for the Weingarten invariants of the lines of curvature are 



14. In the notation of Exercise 13 of Chapter IV, the projective normal joins the 
point x to the point 

2ax vv + (b - a u /2a)x u +(c+a v /2)x v 
when the coordinates x have been multiplied by such a factor that 

&+2c'+(log aVy#%=0 , 2&'-c/a+(log tf/V/PJ.-O , 
where #=a*B /2 +< /2 . ' LANE, 1927. 9 



254 PROJECTIVE DIFFERENTIAL GEOMETRY 

15. Use the method of Section 49 to show that, when the lines of curvature are 
parametric, multiplication of the cartesian coordinates x, y, z, 1 by the function x 
defined by 



converts these coordinates into the coordinates that produce the canonical form 
mentioned in the last exercise for the equations of Exercise 13 in Chapter IV. 

16. Use Exercise 15 to show that, in the notation of Section 48, the local direc- 
tion cosines of the protective normal are proportional to 



17. If an integral surface S of equations (75) with p = q = Q is transformed by 
reciprocal polars, with respect to the sphere x 2 -f y*-\-z z = 1, into a surface S in such a 
way that the tangent plane at a point (x, y, z) of S corresponds to the pole (x, y, z) of 
this plane with respect to the sphere, then the coordinates x, y, z satisfy the equa- 
tions 

== (a 2b)Xu $X v . 

Xuv ~cx axu bx v , 



and we have KK/d*d* = const. Therefore the property of Tzitze*ica is preserved 
under polar reciprocation with respect to the sphere; this sphere can be replaced by 
any quadric with the origin as center. TZITZEICA, 1924. 3, p. 248 

18. Every central quadric has the property of Tzitze*ica. 

TZITZEICA, 1924. 3, p. 251 

19. A ruled surface that is not a quadric and that has the property of Tzitze*ica 
has its two flecnode curves coincident at infinity. TZITZICA, 1924. 3, p. 254 

20. According to the theorem of Koenigs in Section 35, the point y given by equa- 
tion (95) with k 5*0 generates a plane net with equal invariants. Find the differ- 
ential equations of the form (IV, 26) for this net, and prove that it determines a 
sequence of Laplace of period three. WILCZYNSKI, 1914. 3, p. 141 

21. The quadric (55) is the quadric of Lie at a point of a surface in case & 2 , 3 have 
the values given in equations (60) and & 4 has the value given by 



cos 2 A) 
sin A)u/E 1 /* sin A+(kJli cos A) V /G 1 /* cos A , 

where A is an angle such that 

CK 2 -#i) sin 2 A = R 2 , (Ri-R*) cos 2 A = Ri . 

DEMOULIN, 1908. 4, p. 566 



METRIC AND AFFINE APPLICATIONS 255 

22. The centers of all the quadrics of Darboux at a point P(0, 0, 0) of a surface S 
lie on the straight line 



where & 2 , & 3 are given by equations (60). This is the line of intersection of the two 
asymptotic tangent planes through P of the two non-developable ruled surfaces of 
asymptotic tangents circumscribing S along the two asymptotic curves through P. 

23. Investigate the developables and focal surfaces of the congruence generated 
by the line of centers of the quadrics of Darboux at a point P on a surface S (see 
Ex. 22). Prove that the developables intersect $ in a conjugate net, and that the 
foci of the generator separate harmonically the point P and the center of the 
quadric of Lie of S at P. DEMOULIN, 1908. 4 

24. The equations of the transformation of coordinates from the local tetra- 
hedron x, x-i, xi, y in the situation of Section 30 to the local trihedron of Section 
48 are 



CHAPTER VII 
SURFACES AND VARIETIES 

Introduction. This chapter contains some further developments of the 
projective differential geometry of surfaces in hyperspace. Moreover, there 
are certain portions of the theory of varieties in general which are so inti- 
mately connected with the theory of surfaces that it seems appropriate to 
include them here also. Varieties are to be regarded as generalizations of 
curves and surfaces, a curve being a variety of one dimension, while a 
surface is a variety of two dimensions. 

The sections of this chapter fall naturally into two groups, the first group 
including the first three sections and finding its inspiration in the researches 
of Segre. Section 52 is devoted to a study of the neighborhoods of various 
orders of a point on a surface, or on a variety, in hyperspace. Section 53 
contains an investigation of the different forms assumed, in the neighbor- 
hood of a point on a surface, by the curve of intersection of the surface and 
a variable hyperplane through the point. Surfaces in space of five dimen- 
sions are of sufficient interest to occupy the next section. 

In the next group, Section 55 is an introduction to the fundamentals of 
Segre's monumental work on varieties which are the loci of linear spaces. 
An example of such a variety, also studied by Segre, is discussed in the 
following section, namely, the locus of the tangent planes of a surface. 
Section 57 is taken up with a consideration of sets of varieties which are the 
loci of linear spaces with the generating linear spaces in correspondence. 
The developments in this section are mainly due to the author. 

Frequent use will be made of the intersection formula 



This formula* is to be interpreted as stating that two varieties of orders 
p, q respectively and of dimensions h, k in a space S n intersect in a variety 
of order pq and dimensions h+k n, when h+k^n. We shall explain the 
notion of order of a variety in the next section. We may state in passing that 
we shall only speak of order for an algebraic variety, that is, a variety that 
can be defined as the locus of a point whose coordinates satisfy one or more 
algebraic equations. 

52. The neighborhoods of a point on a surface, or on a variety. The pri- 
mary purpose of this section is to define and study systematically the neigh- 

* Bertini, 1923. 2, p. 254. 

256 



52] SURFACES AND VARIETIES 257 

borhoods of various orders of an ordinary point on an analytic surface. The 
neighborhoods of the first, second, and third orders will be considered in 
some detail. A local coordinate system will be introduced for each of these 
neighborhoods, and the equations will be written for the more interesting 
configurations associated with the point of the surface in each case. Among 
these the cone of Del Pezzo is perhaps the most worthy of special mention. 

Although a complete theory of analytic varieties is beyond the scope of 
this book, it nevertheless seems desirable at the end of this section to ex- 
tend to an analytic variety of m dimensions certain of the definitions made 
and results obtained for a surface. A noteworthy result is the formula due 
to Mendel for the dimensionality of the space S(k, r) at a point of a variety 
V m in space S n . 

Let us consider, in a linear space of n dimensions S n , a proper analytic 
surface S whose parametric vector equation is x = x(u, v). The neighbor- 
hood of the first order of an ordinary point P x on the surface S is determined 
geometrically by P x and one point consecutive to P x in each direction from 
P x ; this neighborhood consists of the point P x and the configurations that 
can be associated with P x analytically by means of x and the two first 
partial derivatives x u , x v . Similarly, the neighborhood of the second order 
of the point P x consists of P x and the configurations that can be associated 
with P x by means of the first and second partial derivatives of x with re- 
spect to u and v. In general, the neighborhood of order k of a point P x on a 
surface consists of P x and the configurations that can be associated with P x by 
means of the partial derivatives of x with respect to u and v up to and including 
those of order k. 

Through each point P x of a surface there is a curve C of the family de- 
fined by the differential equation dv Xdw = 0. It will be useful to calculate 
some of the total derivatives of x along the curve C. We find 

(x' = dx/du) , 



where G, H, 7, J are defined by placing 

H = x UU u+3x uuv \+3x uvv \ 2 +x vvv \* . 

T f* l O/M \ I . /] 

/ 3/ttut>"T"^*'ut)t;'*~r "^v 



(2) 



258 PROJECTIVE DIFFERENTIAL GEOMETRY 

We restate here for convenience the definition (see 10) of the space 
S(k, r). The osculating space S(k, r) with respect to an element E r (see 3) 
of a curve at a point P x of a surface S is the ambient of, or linear space of 
least dimensions containing, the osculating space Sk at the point P x of every 
curve on the surface S that passes through P x and has at P x the same element 
E r (r<k). In particular, the osculating space S(k, 0) at a point P x of a sur- 
face S is the ambient of the osculating space Sk at P x of every curve on the 
surface S that passes through P x . It follows that the space S(k, 0) is the 
linear space determined by x and the partial derivatives of x up to and including 
those of order k. The formula for the number of these derivatives shows 
that S(k, 0) is ordinarily a linear space of dimensions k(k+3)/2. Moreover, 
the neighborhood of order k of a point on a surface lies in the space S(k, 0) 
at the point. 

The space S(l, 0) is the tangent plane at a point P x of a surface S; the 
theory of the neighborhood of the first order in this plane is already familiar 
to the reader. The tangent line of a curve C at the point P x is determined 
by x, x'. The locus of the tangent lines at the point P x of all curves on the 
surface $ that pass through P x is a flat pencil with center at P x and lying in 
the tangent plane. If a local coordinate system is introduced in this plane, 
for which the triangle of reference has the vertices x, x u , x v and for which the 
unit point is chosen so that the point x&+ X2X u +x^x v has local coordinates 
proportional to xi, x<t, x 3 , then the equation of the tangent line of the curve 
Cis 

(3) Xz 2 -Z3 = . 

The neighborhood of the second order* of the point P x lies in the space 
S(2j 0), which is ordinarily a space $ 5 . The osculating plane of the curve C 
at the point P x is determined by x, x', x tf . The space S(2, 1) of the surface S 
in the direction of the curve C at the point P x is studied by holding X fixed and 
allowing X' to vary in the formulas for #', x" in the first two of equations 
(1). Under these circumstances x and x' are fixed, but the point x" describes 
the straight line joining the fixed point x v to the fixed point G (see Fig. 40). 
Consequently the osculating plane describes an axial pencil with the tan- 
gent of the curve C as axis and lying in the space 3 of the points 

(4) x, x u , x v , G . 

This space 3 is the space S(2, 1) under consideration. 

When the curve C varies through the point P x so that its tangent line 
at P x assumes all possible directions in the tangent plane, then X varies and 

* Segre, 1907. 2, p. 1050. 



(52] 



SURFACES AND VARIETIES 



259 



the space S(2 t 1) varies, joining always the fixed tangent plane to the vari- 
able point (7, and generating a variety of four dimensions F 4 . Calling the 
locus of a one-parameter family of spaces Sh through a fixed space Sk(h>k) 
a cone with the spaces S h for generators and with the space & for vertex, 
we see that in this extended sense of the word the variety F 4 is a cone whose 
vertex is the tangent plane and whose generators are the spaces S(2, 1). 




FIG. 40 

This cone is called the cone* of Del Pezzo at the point P x of the surface S. 
It can also be characterized as the locus of the 2 osculating planes at the 
point P x of all the curves on the surface S that pass through P x . 

If a local coordinate system is introduced in the space S(2, 0) so that a 
point x&+ +x& vv has local coordinates proportional to x\, . . . , x 6 , 
the equations of the configurations in the neighborhood of the second order 
can be calculated. The equations of the space S(2, 1) can be written in the 
form 

(5) 2Xz 4 -z 6 = 0, Xz 6 -2o; 6 = 0, 

and elimination of X therefrom gives the equation of the cone of Del Pezzo , 



(6) 



* Del Pezzo, 1886. 1, p. 177. 



260 PROJECTIVE DIFFERENTIAL GEOMETRY 

The equations of the locus of the point G are 

(7) xi = #2 = 3 =5 4x^8 = . 

Therefore this locus is a conic in the plane of the points x uv , x uv , x vv , the 
equations of this plane being 

(8) Xi = X2 = X$ = Q 

The neighborhood of the third order of the point P x lies in the space S(3, 0), 
which is ordinarily a space S 9 . The first problem is to study the space 
S(3, 2) associated with a curve C through the point P x on the surface S. 
The osculating space $ 3 of the curve C at the point P x is determined by 
x y x' y x", x"' . When A, X' are fixed and X" varies in the first three of equa- 
tions (1), the points x, x', x 1 ' are fixed and the point x'" varies on the line 
joining the point x v to the fixed point H-\-3(x uv +x vv \)\'. Calling the locus 
of a linear one-parameter family of spaces Sk~i through a fixed space Sk-z 
in a space /S* a pencil with the space &_ 2 for base, we see that, under the 
conditions stated, the osculating space 83 of the curve C varies and has for 
its locus a pencil with the osculating plane as base and lying in the space S* 
of the points 

(9) x, x u> x v , G, H+3(x uv +x vv \)\' . 

This space 4 is the space 5(3, 2) under consideration; it contains the space 
$(2, 1) previously studied. 

The next problem is to investigate the space 5(3, 1) in the direction of the 
curve C. This problem is the same as to investigate the locus of the space 
5(3, 2) when X' varies and X is fixed. Under these circumstances the space 
5(3, 2) varies and has for its locus a pencil with the space 5(2, 1) as base and 
lying in the space 5 6 of the points 

(10) x, x u , x v , x uu +x uv \ x uv +x vv \ H . 

This space 5 6 is the space 5(3, 1) under consideration, and contains the tan- 
gent plane of the surface S at the point P x . 

Finally, the locus of the space 5(3, 1) when the curve C varies through the 
point P X) so that its tangent line at P x assumes all possible directions in the 
tangent plane, is a variety of six dimensions 7 6 . This variety is a cone whose 
vertex is the tangent plane and whose generators are the spaces $(3, 1). 

The order of an algebraic variety V m of dimensions m in space S n is defined 
to be the number of points in which it is met by any linear space 5 n _m 



52] SURFACES AND VARIETIES 261 

which is not a component of the variety V m . In discussing the order of a 
cone the linear space whose intersection with the cone is used must have no 
point in common with the vertex of the cone. So, for example, the cone of Del 
Pezzo is of order two, since the plane of the points x uu , x uv , x vv meets it in a 
conic, which is certainly of order two. The symbol for the cone of Del 
Pezzo is VI, the subscript indicating the dimensions and the superscript the 
order of the cone. 

For the purpose of showing that the cone V 6 defined above is of order five, 
so that it may be denoted by the symbol VI, let us observe that the order 
of this cone is the order of the locus of the plane of the last three of the 
points (10) when X varies. This locus is a variety V$ in the space S Q of the 
points x uu , . . . , x vvv . Now, when X varies, the locus of the point x uu +x uv \ is 
obviously the straight line joining the points x uu , x uv ; similarly, the locus of 
the point x uv +x vv \ is a straight line ; and the locus of the point H is a twisted 
cubic curve in the space $ 3 of the points x uuu , . . . , x vvv . Moreover, the 
three points where the variable plane meets the two straight lines and 
the cubic curve are projectively related, all three points corresponding to 
the same value of X. Therefore the order of the locus of the plane is five, 
being the sum of the orders of the director curves. 

For some purposes it is advantageous to introduce a local coordinate 
system in the space $(3, 0) so that a point x\x+ -\-x^x vvv has local 
coordinates proportional to xi, . . . , x w . Then the equations of the space 
$(3, 2) can be written in the form 



The equations of the space S(3, 1) are found, by eliminating X' from equa- 
tions (11), to be 

rX 2 z 4 - 



Finally, the equations of the cone V\ are found, by eliminating X from equa- 
tions (12), to be 

(13) a|337Z9 = , z 2 , 3^8^10 = , x^ x 6 x 9 -{-3xQX 7 = . 

It may be remarked that the cone V\ is only part of the intersection of the 
three hyperquadrics whose equations appear here, the space $ 6 with the 
equations z 9 = # 8 *= 6 = 0, and the space *S 6 with the equations x 9 = z 8 = x 7 = 
counted twice, being excluded. 



262 PROJECTIVE DIFFERENTIAL GEOMETRY 

We shall not prolong further the detailed discussion of particular neigh- 
borhoods of a point P x on a surface S, but shall state without proof* two or 
three results concerning the general neighborhood of order k. For this pur- 
pose let us denote by V(k, r) the locus of the osculating space & at the point 
P x of every curve on the surface S through P x having at P x the same ele- 
ment E r . The space V(k, r) is of dimensions 2k r. The space S(k, r) is of 
dimensions 

(14) [fc(/c+r+3)+c(r-l-c)]/2(r+l) , 

where 

0<!c^r , k=c (mod r+1) . 

The space V(k, r) is linear when r^ (k l)/2. 

Although this book is not primarily concerned with varieties in general, 
it seems well to define a general analytic variety and to state a few results 
concerning the general neighborhood of a point thereon. An analytic variety 
V m in space S n is defined to be the locus of a point whose coordinates x are ana- 
lytic functions ofm (and not fewer than m) independent parameters u l , . . . , u m 
(m^n). The parametric vector equation of the variety V m can be written 
in the form 

(15) x = x(u l , . . . , u m ) . 

The tangent linear space S m at a point P x of a variety V m is defined to be the 
ambient of the tangent line at P x of every curve on V m that passes through P x . 
Such a curve can be represented by equations of the form 

u l = u l (f) (t = l, . . . , ra) , 

where t is an independent variable. The tangent space S m is easily shown to 
be determined by the points x, xi, . . . , x m , subscripts indicating partial 
differentiation with respect to the corresponding variables w 1 , . . . , u m . 
This space S m is the space S(l, 0) at the point P x of the variety V m . 

The definitions of the spaces S(k, 0), V(k, r), S(k, r) as made in connec- 
tion with surfaces can easily be restated so as to be available in the theory 
of a general variety V m . We shall not enter into details but shall state a 
few results without proof. The space S(k, 0) is of dimensions C m +k,k l, the 
number of combinations of m+k things taken k at a time being denoted by 

* Mendel, 1930. 3. The proof may be found in Mendel's Chicago doctoral dissertation, 
1930. 



53] SURFACES AND VARIETIES 263 

Ck+m, k. The space V(k, r) is of dimensions m(kr)+r. The space S(k, r) is* 
of dimensions 

(16) (r+l)C m+p . ltm +(c+l)C m+p - ltP -l , 

where 

O^c^r , fc = p(r+l)+c, 

and k, r, c, p are positive integers or zero. 

63. Hyperplane sections of a surface. The intersection of a surface and 
a hyper plane is a curve. A variable hyper plane through a fixed point of a 
surface intersects the surface in a curve which may assume various forms in 
the neighborhood of the point according to the different possible relative 
positions of the surface and hyper plane. Some of the possibilities as to the 
nature of the curve will now be considered, the surface being unspecialized. 
The investigation will be continued in the special case in which the surface 
is an integral surface of a differential equation of the second order. 

The equation of a hyperplane iS n _i in a space S n can be written with cus- 
tomary abbreviation in the form S x = 0, the summation ranging from 1 to 
n+l. The n-\-l coefficients are homogeneous coordinates of the hyper- 
plane. Let us consider a surface S with the parametric vector equation 
x x(Uj v) in the space S n . The coordinates X of a point near an ordinary 
point P x on the surface S can be represented by Taylor's expansion as 
power series in the increments Aw, At; corresponding to displacement on S 
from P x to the point X. If the hyperplane passes through the point X, we 
have 2 %X = and consequently 



(17) 2&+2& 1 &u+2&i&v+(2!;Xn&u i + )/2H ---- =0 , 

the subscripts 1, 2 now denoting partial derivatives with respect to u, v 
respectively. If the hyperplane passes through the point P x , then Sc = 0. 
Such a hyperplane ordinarily intersects the surface S in a curve with a 
simple point at P x ; the direction of this curve at P x is found, by means of 
equation (17), to be given by 

(18) 2fcridtt+S&r 2 eto = . 

A hyperplane which contains the tangent plane at the point P x of the 
surface S has coordinates which satisfy the conditions 

(19) Sr = 2ci = 2r 2 = 0. 

* Ibid. For the proof see Mendel's dissertation previously cited. 



264 PROJECTIVE DIFFERENTIAL GEOMETRY 

Such a hyperplane ordinarily intersects the surface S in a curve with a 
double point at P x ; the directions of the double-point tangents are given by 

(20) Z&ndu^Sfrududv+Z&Mdv* = . 

The curve of section of the surface S made by a hyperplane through the 
tangent plane has two coincident double-point tangents at the point P x if, 
and only if, the coordinates of the hyperplane satisfy not only equations 
(19) but also 

(21) 2& n S& 22 -(S&i 2 ) 2 = 0. 

The double point is then ordinarily a cusp. In this case the direction of the 
cusp tangent satisfies the two equivalent equations 

(22) 2&ndu+2&udv = , Z&izdu+Z&wdv = . 

It is interesting in this connection to consider the equations of certain 
already familiar configurations in hyperplane coordinates. A conic meets a 
hyperplane in two points. Replacing X by dv/du in the first of equations 
(2) and calculating the expression S(r, we see that the conic which is the 
locus of the point G as X varies meets any hyperplane in two points which 
are obtained by solving equation (20) for the ratio dv/du. These two points 
coincide if, and only if, equation (21) is satisfied. Then the hyperplane con- 
tains a tangent of the conic and may be said to be itself tangent to the 
conic. Thus (21) is the equation of the said conic in hyperplane coordinates. 
Moreover, equations (19), (21) taken together are the equations of the cone 
of Del Pezzo in hyperplane coordinates. 

A hyperplane which contains the space (2, 0) at the point P x of the 
surface S has coordinates which satisfy not only the conditions (19) but 
also 

(23) Sz u = Sz 12 = ;Sz 22 = 0. 

Such a hyperplane ordinarily cuts S in a curve with a triple point at P x ; the 
directions of the triple-point tangents are given by 

(24) 2&mdu 3 +3SfcW^ . 

Various possibilities can arise with regard to the coincidences of these 
tangents, but we shall not go into the details here. Moreover, we shall 
leave to the reader the discussion of the curve of intersection of a surface 



53] SURFACES AND VARIETIES 265 

and a hyperplane through the space S(k, 0) at a point of the surface with 
fc>2, and with k general. 

An integral surface of just one second-order differential equation of the form 
(IV, 10) will be called for brevity a surface F. Such a surface has many 
special properties, some of which have already been observed in Chapter IV. 
For example, such a surface sustains a conjugate net or else a one-parameter 
family of asymptotic curves. Moreover, the space S(2, 0) at a point P x of a 
surface F is a space S 4 . 

There are some special properties of a surface F which are connected with 
the hyperplane sections of the surface. For example, the pairs of double-point 
tangents, at a point P x on a surface F, of the curves of section of F and the hyper- 
planes containing the tangent plane of F at P x are the < l pairs of rays in the 
involution* having the conjugate tangents (supposed distinct) at P x for self- 
corresponding rays. In order to demonstrate the truth of this statement let 
us observe that the directions of the double-point tangents are given by 
equation (20), and the directions of the conjugate tangents by 

(25) Cdu* - 2Bdudv+Adv z = . 

Moreover, equations (19) and (IV, 10) imply 
(26) 



But this equation says that the harmonic invariant of the binary quadratic 
forms in equations (20) and (25) vanishes. Thus the proof is completed. 

It will next be shown that the conjugate directions at a point of a surface F 
are the hessian pair of the triple of directions given by equation (24). Use will 
be made of the property of the hessian pair of a triple which is set forth in 
Exercise 6. Let us for convenience write equations (19), (23) together in 
the concise form 

(27) Zfcc = , S$c p = , 2fy pq = (p, q = 1, 2) . 

It is easy to deduce from (27) and the equations obtained therefrom by 
differentiation the relations 



(28) 2^ = 0, Zs{p = 0, 2a M = 0, 
Moreover, continuation of the calculations leads to 

(29) 2trX pq = - Zfr rpq = 2xt rpq = - 2Xrl- pq (p, q, r = 1, 2) . 
* Segre, 1907. 2, p. 1063. 



266 PROJECTIVE DIFFERENTIAL GEOMETRY 

From equation (IV, 10) one now obtains 

A2S pzn-f 2SpZi2+CS p Z22 = (p = 1, 2) , 

and consequently 

(30) A2txn P +2B2tx 12p +C2&M p = Q (p = 1, 2) . 

The equation of the first polar pair of an arbitrary direction 8v/5u with re- 
spect to the triple of directions (24) is 



, . 

{ } 



\ 



The demonstration is completed by observing that equation (30) says that 
the harmonic invariant of the binary quadratic form in equation (31), in 
the differentials du, dv, and of the form in equation (25) in the same vari- 
ables vanishes identically in 8u, 8v. 

It is clear geometrically that the conjugate tangents at a point of a surface F 
are the cusp tangents of the curves of section made by those hyperplanes through 
the point that cut the surface F in curves with cusps at the point. Moreover, at 
a point of a surface F, the hyperplane envelope represented by equation 
(21) breaks up into two spaces S 3 within the space S(2, 0), as can be verified 
by use of equations (21), (26). The sections with cusps are made by the 
hyperplanes through one or the other of these two spaces 83. 

54. Surfaces immersed in space S 5 . There are certain special properties 
of surfaces immersed in space $ 5 which are of considerable interest. For 
instance, there are on such a surface five noteworthy co variant one-pa- 
rameter families of curves called principal curves. In this section these 
curves will be considered first on the most general analytic surface and then 
on a surface F in space $5. Investigation of those surfaces on which the 
principal curves are indeterminate will introduce the surface of Veronese. 

Let us consider some of the possible relative positions of a hyperplane 
$4 and a surface in space $ 5 . There are in all > 5 hyperplanes in space 5 . 
Among them there are < 2 hyperplanes containing the tangent plane at a 
point P x of a surface S; the coordinates of any one of the latter satisfy 
the three equations (19). Among these latter there are oo 1 hyperplanes that 
cut the surface S in curves with cusps at the point P x ; the coordinates of any 
one of these oo 1 hyperplanes satisfy also equation (21), or what is the same 
thing, the two equations (22). It is easy to show that among these hyper- 
planes there are just five that cut the surface S in curves with tacnodes at 
the point P x . In fact, the coordinates of any such hyperplane must satisfy 



54] SURFACES AND VARIETIES 267 

also equation (24). Elimination of the six homogeneous coordinates from 
the six equations (19), (22), (24) leads to the equation 



(32) { 



(x, xi, z 2 ) Xudu+Xudv, 



This equation is of the fifth degree in the ratio dv/du, and consequently 
there are five hyperplanes which produce tacnodal sections, as stated. 

We now state some definitions. The five directions of tacnodal section at a 
point of a surface in space S$ are called the principal directions at the point, 
and the tangents in these directions are called principal tangents. The five one- 
parameter families of curves enveloped by the principal tangents are called the 
principal curves on the surface. Equation (32) may be regarded as the 
curvilinear differential equation of the principal curves. 

The coordinates x of a general point on a surface which is non-develop- 
able and not a surface F and is immersed in space 5 satisfy* a system of 
four equations expressing each of the third derivatives of a; as a linear combi- 
nation of x, x u , x v , x uu , Xuv, x vv . If non-homogeneous coordinates are used, 
and if two of them are taken as the independent variables u, v, the system 
of differential equations can be written in the form 



(33) 



Xuuu ax uu -f- rx uv -f- 5x vv 

Xuuv == OX U u~T~SX-u V -p 'yXw 
Xuvv = CX UU ~T~ffX uv -\-pX vv 

x vvv = dx uu +px uv +ax vv 



In this notation the curvilinear differential equation (32) of the principal 
curves becomes 



(34) { 



5du 5 +(3y-r)du*dv+(a-3s+3p)duW 



In space $ 5 integral surfaces F of a differential equation of the form 
(IV, 10) are of particular interest. For example, one can easily demonstrate 
the following theorem. Through each point P x of a surface F in space S$ there 
is a unique hyperplane S* which cuts F in a curve with a triple point at P x ; this 
hyperplane is the space S(2, 0) at P x . For the demonstration one observes 
that the six coordinates of any hyperplane cutting the surface F in a curve 
with a triple point at a point P x satisfy the six equations (19), (23). But 

* Beenken, 1928. 6, p. 2. 



268 PROJECTIVE DIFFERENTIAL GEOMETRY 

just five of these equations are independent in the present situation, since 
account must be taken of equation (26). These five equations can be solved 
uniquely for the ratios of the coordinates , and then the unique hyperplane 
is observed to be the space /S(2, 0), as asserted. The directions of the 
triple-point tangents of the curve of section of the surface F made by the 
space S(2, 0) at the point P x are given by equation (24). 

It will now be shown that the five principal directions at a point P x of a 
surface F in space 85 consist* of the two conjugate directions and the three 
directions of the curve of intersection of the surface F and the space S(2, 0) at 
the point P x . In addition to the equations previously used in studying the 
principal directions, equation (26) is now valid. There are two possibilities 
with respect to equations (26) and (22) regarded as simultaneous linear 
homogeneous algebraic equations in the expressions S&cn, #12, 2r22. 
First, it may be that the hyperplane is such that these expressions are not 
all zero. Then the determinant of their coefficients must vanish. The 
vanishing of this determinant leads to equation (25). Therefore the con- 
jugate directions are two of the principal directions. Second, if the expressions 
considered are all zero, equations (23) being valid, the hyperplane is the 
space *S(2, 0) at the point P x , and the principal directions are given by 
equation (24). This completes the proof. 

The question arises as to what surfaces are such that on them the prin- 
cipal curves are indeterminate. It can easily be shown that the principal 
curves are not indeterminate on a surface F immersed in space S$. For, in the 
first place, since not all of A, B, C are zero, the conjugate net, or the family 
of asymptotics as the case may be, cannot be indeterminate. In the second 
place, it can be shown by an indirect proof that the curves defined on a 
surface F by equations (24) cannot be indeterminate. For, if they were, 
then we should have 

(35) S&W = (p,<?,r=l, 2). 

If the surface F sustains a conjugate net, and if this net is taken as para- 
metric, then the coordinates of the hyperplane 8(2, 0) can be written in 
the form 

(36) {=(3,31,32,311,322) . 

Equations (35), (36) imply 

(37) (3, 3i, 3 2 , 3n, 3 22 , X pqr ) = . 

* Segre, 1921. 1, p. 203. 



55] SURFACES AND VARIETIES 269 

Therefore each third derivative of x can be expressed as a linear combina- 
tion of x, Xij x<i, Xn, #22, and the surface F is in a space $ 4 . Thus the assump- 
tion that the surface F is immersed in space S 6 is contradicted. If instead 
the surface F sustains a family of asymptotics which are taken as the t>- 
curves, then the coordinates { of the hyper plane $(2, 0) can be written in 
the form 

(38) {=(, a*, z 2 , xu, 12) , 

and the same contradiction can be reached. 

It can be shown that the principal curves are indeterminate on a developable 
surface. For, according to Exercise 3 of Chapter IV, the coordinates x of a 
general point on a developable surface satisfy two independent equations 
of the form (IV, 10). Then it is possible to eliminate two of the second 
derivatives from equation (32) ; consequently the first five columns of the 
determinant in (32) are linearly dependent; hence equation (32) is an 
identity in dv, du. 

We now state a definition. A surface of Veronese is a surface* immersed 
in space $ 5 and sustaining a two-parameter family of conies. The algebraic 
equations of a surface of Veronese can be written in the form 

(39) Ziz 5 = z 4 z e , 2^6 = x&s , x 3 X4=x 6 Xs , 
and a parametric representation of this surface is 

(40) Xi u^ , #2=1, x$ = v 2 , x\ u , x 6 v , x 6 = uv . 
Since these coordinates x satisfy the differential equations 

(41) OW = (p, ?, r=l, 2), 

it follows that the principal curves are indeterminate on a surface of Veronese 
(see Exs. 18, 19). 

55. Varieties which are the loci of linear spaces. In this and the next two 
sections attention will be focused on varieties f which are the loci of linear 
spaces. In the present section a variety Vk+ 1 which is the locus of < i linear 
spaces Sk in space S n is briefly studied. The most interesting special case 
is perhaps the case fc = l, in which a variety Vk+i is a ruled surface. Some 
of the results obtained for a variety V k +i are employed in a generalization 

* Bertini, 1923. 2, p. 394. 
t Segre, 1910. 1. 



270 PROJECTIVE DIFFERENTIAL GEOMETRY 

of the theory, in which one considers a variety Vk+ m which is the locus of 
oo m linear spaces &. 

The word "consecutive" is unfortunately used in two senses in differen- 
tial geometry. The first is connected with a limiting process and is the sense 
in which the word has been used hitherto in this book. In this sense of the 
word the usual definition of the tangent at an ordinary point of a curve is 
succinctly stated by saying that the tangent is the line that joins the point 
to a consecutive point on the curve. Moreover, three consecutive points 
determine an osculating plane of a curve, and four consecutive points, or 
else two consecutive tangent lines, determine an osculating space 83 of the 
curve. Finally, according to Exercise 1, two consecutive tangent planes of 
a surface determine a space $ 6 . The second sense of the word is connected 
with the order of an infinitesimal, and when the word is used in this sense in 
the following pages it will be italicized. According to this sense, two con- 
secutive generators of a developable surface intersect in a point, so that two 
consecutive tangents of a curve determine an osculating plane of the curve. 
Moreover, two consecutive tangent planes of a surface have a point in com- 
mon and hence determine a space S*. 

A variety Vk+i which is the locus of oo 1 linear spaces Sk in space S n (k< 
n 1) can be represented analytically in the following way. A space Sk 
can be regarded as determined by k+ 1 linearly independent points xi, . . . , 

i with the coordinates 



(42) x{ (i=l,... 

Let all the coordinates of all the k+l points be single- valued analytic func- 
tions of the same independent variable t. Then, as t varies, the locus of each 
point Xi is a curve C, and the locus of the space Sk is a variety Vk+i. Any 
point y on this variety is given by the equation 

(43) 2/ = Sw t z t (i = l, . . . ,fc+l) , 

in which the coefficients HI, . . . , Uk+i are parameters which are independent 
of t. Therefore equation (43) is the parametric vector equation of the variety 

n +1 . 

The curves d may be called director curves of the variety Vk+i- The 
points Xi of these curves are in correspondence, a set of corresponding points 
consisting of those that correspond to a particular value of the parameter t. 
Each set of corresponding points determines a generator S k , and on this 
generator the parameters u t are protective homogeneous coordinates of the 
point y. 



55] SURFACES AND VARIETIES 271 

Any curve C through a point P y on a variety V k +i may be represented by 
equations of the form 

(44) Ui = Ui(t) (t-1, ...,*+!), 

in which t is an independent variable. The tangent line of the curve C at the 
point P v is determined by P v and the point y f given by 

(45) y' = 2u'x+z, 

where the accents denote differentiation with respect to t, and z is defined by 

(46) z 



The tangent space Sk+i at a point P v of a variety Vk+i is by definition the 
ambient of the tangent line at P v of every curve on Vk+i that passes through 
P v . This tangent space can be shown to be determined by the points 

(47) i,..., Xk+i, ' z , 

either by applying the result in the latter part of Section 52 on the determi- 
nation of the tangent space S m at a point of a variety V m , or else by observ- 
ing that when the point P v is fixed then t and u* are fixed and that when the 
curve C through P v varies in all possible ways then the derivatives u( vary, 
so that the point y f varies in the space Sk+i. 

The tangent space Szk+i along a generator Sk of a variety Vk+i is defined to be 
the ambient of the tangent space Sk+i at a point P v on the generator Sk as P v 
varies over Sk. Holding t fixed and allowing u % to vary, we see that the point 
z describes the* space of the points zj, . . . , x'k+i. Therefore the tangent 
space Stk+i is determined by the 2&-J-2 points x\, . . . , Xk+i, #(,..., x'k+i. 
It has been tacitly assumed that these points are linearly independent, and 
that 2&+1 ^n. If there should happen to be h independent linear relations 
among the 2k+2 points, then we should speak of the tangent space SU+I-H 
along the generator Sk of the variety Vk+i. 

The point y given by equation (43) on a generator Sk of a variety Vk+i 
will be said to lie also on a consecutive generator in case there exist parame- 
ters w % such that 

(48) y^UiXi^u^+w^^+xW) '(i = l, . . . , fc+1) , 

the generator consecutive to Sk being regarded as determined by the points 
Xt+xldt. Such a point y is called a singular point, or focus, of the generator 



272 PROJECTIVE DIFFERENTIAL GEOMETRY 

Sk t and the locus of the foci on Sk is called the singular space of this generator. 
Equations (46), (48) imply 

(49) z+2wr = 0, 

infinitesimals of higher order than the first being neglected. Therefore the 
point z corresponding to a singular point y lies on the generator Sk contain- 
ing the point y, and the tangent space Sk+i at a focal point of a variety Vk+i is 
indeterminate. Moreover, if there are h independent foci on a generator Sk, 
the linear space tangent to the variety Vk+i along Sk is a space Sw+i-h called 
the singular tangent space along the generator & of the variety V* +1 . If the 
parameters wt are functions of t, a focal point y describes a curve when t 
varies, and the tangent at a point of this curve lies in the generator Sk 
through the point. 

The foregoing considerations can be generalized in the following way. 
Let us consider a variety Vk+ m which is the locus of o> TO linear spaces 
Sk(k<nm). Such a variety Vk+ m can be represented analytically as be- 
fore, except that the coordinates of the points x\, . . . , Xk+i are now func- 
tions of m independent variables t 1 , . . . ,t m . Any point y on the variety 
Vk+m is still given by an equation of the form (43), but the locus of each 
point Xi is now naturally called a director variety V m . 

The differential geometry of a variety Vk+ m , in the neighborhood of a 
generator Sk, can be studied by considering a variety Vk+i within the 
variety Vk+m and through the generator Sk, so that the preceding results 
can be applied. Such a variety Vk+i can be defined by placing 

(50) t> = t>(r) O'=l, ...,m), 

where T is an independent variable which takes the place of t in the applica- 
tion of the preceding results. The point z, corresponding to a point y on 
the generator Sk and defined by equation (46), is now given by 

(51) z = IEui(dXi/dV)(dtfldT) (i=l, . . . , fc+1; j=l, . . . , m) . 

The tangent space Sk+i at the point y on the generator Sk of the variety Vk+i is 
determined, as before, by the points Xi, . . . , Xk+i, z. The tangent space 
SM+I along the generator Sk of the variety Vk+i is easily shown to be deter- 
mined by the points 



(52) {*" ' ' ' ' 



1, . . . , m) 



56] SURFACES AND VARIETIES 273 

It may be noted here that fixing r fixes the generator S k ; the values of w 
associated with this value of r determine the point y on S k ; the form of the 
functions t*(r) determines the variety V k +i within the variety V k + m ; and 
when the variety V k+ i varies in all possible ways through the fixed genera- 
tor Sk, the derivatives dt'/dr, for the fixed value of r, vary. 

The tangent space S k + m at a point P v of the variety V k+m is defined in the 
usual way and is determined by the points 



, dy/dt* (i=l, . . . , fc+1; j = l, . . . , m) , 

of which only k+m+l are linearly independent. Indeed, this space S k +m 
is found to be determined by the fc+ra+l points 

(53) x t , 2u t (dx t /dfi), . . . , 2Ui(dXi/df*) (i = l, . . . , M-l) . 

Moreover, the tangent space S k + m could have been defined as the ambient of the 
tangent space S k +i at the point P y of the variety V k +i as V k+ i varies in all pos- 
sible ways through the generator S k containing P y . The tangent space S k+m 
projects from the generator Sk the space S m -i of the points 

2u t (dx t /dfi), . . . , 2ut(dXi/dt) (i = l, . . . , fc+1) , 

which is the locus of the point z when the variety V k +i varies. 

The tangent space S(k+i)(m+i)-i along a generator S k of the variety Vk+m 
is defined to be the ambient of the tangent space Su+i along the generator S k of a 
variety V k +i through S k , as the variety V k +i varies in all possible ways through 
the generator S k and within the variety V k + m . The tangent space $(jfc+i)(m+i)-i 
may also be characterized as the ambient of the tangent space Sk+m at a 
point y on the* generator Sk of the variety V k+m as the point y varies over the 
generator &, and is determined by the (fc+l)(w+l) points 

(54) x t , dXi/dt* (i = l, . . . ,M-1;; = 1, . . . , m) . 

The dimensions of the tangent space along a generator Sk of a variety V k+m 
would be less than (fc+l)(w+l) lif there were one or more foci on each 
generator S k . 

56. The locus of the tangent planes of a surface. An interesting illus- 
tration of the theory in the latter part of the last section is found in the 
special case of the locus of the tangent planes of a surface immersed in 
space S n (n>4). Since a plane is a space S 2 , and since there are co* points 
on a surface, we place k=m=2 in the theory of a variety Vk+m and observe 
at once that the locus of the tangent planes of a surface is ordinarily a variety 



274 PROJECTIVE DIFFERENTIAL GEOMETRY 

V*. If the parametric vector equation of the surface is x = x(u, v), then the 
parametric vector equation of the variety F 4 can be written in the form 

(55) y = x+8x*+tx 9 , 

where s, t, u, v are independent variables. For the purpose of specializing 
the preceding discussion we place u\ = 1 , u* = s, u$ = t; x\ = x, x% = x u , z 3 = x v ; 
P = u, P = v. 

Let us notice the special cases in which the locus of the tangent planes of 
a surface has dimensions less than four, and then exclude these cases from 
further consideration. Of course, if the surface were a plane, the locus would 
be simply this plane. Moreover, if the surface were a developable not a 
plane, or if it were a non-developable surface in space 3 , the locus of its 
tangent planes would be a variety F 3 . The converse of the last statement 
is also true. For, if the locus is a variety F 3 , the matrix of five columns and 
n+l rows made of y, y s , y t) y u , y v is of rank four. Differentiation gives 

(56) y 8 = x u , yt = x vy y u = x u +sx uu -\-tx uv , yv = x v +sx uv +tx vv . 

Hence the vanishing of a general one of the fifth order determinants of the 
matrix may be expressed by the equation 

\X) Xu) Xi)j SXuu~\ vXuv) SXui)~\~lXi)vj == U , 

which is an identity in s, t, u, v. Using the fact that it is an identity in s, t } 
one sees that x satisfies two independent equations of the second order of 
the form (IV, 10). Therefore the surface is developable or else is a non- 
developable surface immersed in space /S 3 , as was to be proved. 

The tangent space 4 at a point y of the variety F 4 associated with a 
surface S is determined by the points y, y s , y t , y u , y v , now supposed to be 
linearly independent. Since this space $ 4 depends not on s, t separately but 
only on the ratio s/t, it follows that at every point on a tangent line at a point 
x of the surface S the tangent space >S 4 of the variety F 4 is the same. Therefore 
the variety F 4 has at most 3 tangent spaces S*. 

It is easy to show that two consecutive tangent planes of a surface have 
at least one point in common. In fact, the two tangent planes may be re- 
garded as determined by the two sets of three points 



(57) *' 



x+x u du+x v dv , 



57) SURFACES AND VARIETIES 275 

The point x+x u du+x v dv is obviously in both planes. Consequently, the 
contact point of a tangent plane is a focus of the plane regarded as a generator 
of the variety F 4 . 

The tangent planes of a surface S constructed at the points of a curve 

(58) u = u(r) , v = v(r) 

on S form a variety Vs within the variety F 4 . The tangent space $ 3 at a 
point y of this variety F 3 is determined by the points x, x u , x v , z, where z 
is given, according to equation (51), by 

(59) z = x u u'+x v v'+s(x uu u'+x uv v')+t(x uv u'+x vv v') , 

accents indicating differentiation with respect to T. Since each tangent 
plane of the surface S has a focus, the linear space tangent to the variety V 3 
along a tangent plane of the surface S is a space $ 4 instead of the space $ 5 
that might have been expected. This space S* is determined, according to 
(52), by the points 

V^OU) X, Xu, Xy, XutiU \~XuvV , XuvU ~| XyyV . 

The tangent linear space along a generator of the variety F 4 , i.e., along a 
tangent plane of the surface S, is ordinarily a space S 5 instead of the space 
Ss that might have been expected. This tangent space S& is determined, 
according to (54), by the points x, x u , x v , x uu , x uv , x vv , and is identical with 
the space S(2, 0) at the point x of the surface S. 

If the fundamental surface is a surface F, the tangent space $ 4 at a point y 
of the variety F 4 is independent of the ratio s/t, as can easily be verified by use 
of equations (56) . Conversely, if the space $ 4 is independent of s/t, the surface 
is a surface F, since, on setting s = 0, t = 1 in equations (56), one sees that the 
space of the points x, x u , x v , x uv , x vv must be the same as the space of the 
points x, x u , x v , x uu , x uv corresponding to s = 1, t = 0; therefore the surface is a 
surface F. Consequently, the tangent space $ 4 at a point of the variety F 4 
which is the locus of the tangent planes of a surface F is the same at all points of 
a tangent plane of F, so that this variety F 4 has only o 2 tangent spaces $ 4 , 
instead of the usual o 3 (see Ex. 16). Moreover, the tangent space along a 
generator of the variety F 4 (ordinarily a space $ 5 ) is, for a surface F, this 
same space 4 , and is the space S(2, 0) as in the case of an unspecialized sur- 
face. 

67. Sets of varieties which are loci of linear spaces, with the generators 
in correspondence. The projective differential geometry of certain sets of 
varieties in space S n , which are loci of linear spaces and which have their 



276 PROJECTIVE DIFFERENTIAL GEOMETRY 

generators in correspondence, can be studied effectively by means of systems 
of linear homogeneous differential equations of the first order. When each 
variety is the locus of w 1 linear spaces, the differential equations are ordi- 
nary; otherwise partial differential equations are used. 

This section will be concerned for the most part with the protective 
differential geometry that can be studied by means of a system of n+1 
ordinary linear homogeneous differential equations of the first order in n+1 
dependent variables. In the latter part of the section a system of ra(n+l) 
linear homogeneous partial differential equations of the first order in n+1 
dependent variables and m independent variables will be introduced. 

In space S n let us consider n+1 linearly independent points xi, . . . , x n +\ 
with projective homogeneous coordinates 

(61) 4 (i,j=l,...,n+l), 

and let us first suppose that these coordinates are single-valued analytic 
functions of one independent variable t. Then, as t varies, the locus of each 
point Xi is a curve C, and we thus obtain n+1 curves with their points in 
correspondence, corresponding points being those that correspond to the 
same value of the parameter t. 

The square matrix of the coordinates x{ of the points x t is of order and 
rank n+1. Therefore it is possible to determine* the coefficients of a system 
of differential equations of the form 

n+1 

(62) *J= 
so that 



will be n+1 sets of solutions. For example, if each of these sets is substi- 
tuted in turn in the first equation of system (62) with its n+1 coefficients 
a\t regarded as unknown, the resulting n+1 linear algebraic equations can 
be solved uniquely for these coefficients. Similarly, the coefficients of the 
other n equations can be determined. 

The transformation of dependent variables, 



(63) a?,- = X,Si (i = l, . . . , n+1; X scalar^O) 
and the transformation of independent variable, 

(64) w = tt(0 ( 
* Lane, 1928. 5, p. 786. 



57] 



SURFACES AND VARIETIES 



277 



leave each of the curves C invariant. We reach thus the following conclu- 
sion. 

The projective differential geometry of a set of n+l curves in space S n with 
their points in correspondence can be studied by means of the invariants and 
covariants of system (62) under the total transformation (63), (64). 

With a different choice of the transformation of dependent variables the 
geometry of a different configuration can be studied by means of the corre- 
sponding invariants and covariants of system (62). For example, if n is 
odd and if the transformation 



(65) 



4 

2^ 
j-3 



n+l 



(1=1,2), 



(fc-3',4), 



is used instead of the transformation (63), the configuration composed of 
(n+l)/2 ruled surfaces with their generators in correspondence can be 
studied. In general, the number of configurations in space S n whose projective 
differential geometry can be studied by means of the invariants and covariants 
of system (62) under linear transformations of the dependent variables and the 
transformation (64) of the independent variable is n(n 2 -f 20)/24 if n is even, 
and is n(n 2 +23)/24 if n is odd. These formulas can be obtained by observ- 
ing that the number under consideration is the number of ways in which it 
is possible to choose at least two transformations of the form 



(66) 



j-a+l 



(1=0+1,..., A), 



3=0+1 



278 PROJECTIVE DIFFERENTIAL GEOMETRY 

where a, &,..., g, h are positive integers such that 
a^ . . . ^hg , a<6< . . . 



Certain curves called inter sector curves will now be defined and studied. 
Let us consider any one of the possible configurations just indicated in 
space S n and any one of the varieties V k +i in this configuration, which is 
the locus of a space Sk with I<!fc5 ft 2. A curve on such a variety Vk+i 
wilt be catted an intersector curve with respect to the remaining varieties in the 
configuration in case the tangent to the curve at the point where it crosses each 
generator Sk intersects the space S n -k-i determined by the corresponding genera- 
tors of the other varieties. In order to obtain the differential equation of 
these curves let us observe that any curve on the variety Vk+i, with certain 
non-essential exceptions, can be regarded as the locus of the point P y de- 
fined by 

Jb+l 

(67) l/ 



in which the non-homogeneous parameters w 2 , . . . , Uk+i are functions of 
the independent variable t. The point y'+hy is any point (except P y ) on 
the tangent of this curve at P v ; by means of equations (62), y'-\-hy can be 
expressed as a linear combination of xi, . . . , x n +\. The curve is an inter- 
sector curve if the coefficients of x\, . . . , Xk+i in this expression vanish. 
Thus one obtains fc-f- 1 equations, and the elimination of h therefrom results 
in the differential equations of the intersector curves, 

k+i 
(68) Up+ai p anu p + ^ u } (aj p u p aji)=0 (p = 2, . . . , k+l) . 



There are <x> k of these curves, one of them passing through each point of 
each generator &. 

The tangents of the intersector curves may be called intersector tangents. 
The locus of the intersector tangents at the points of a fixed generator Sk 
is a variety Vk+i whose local equations can be found in the following way. 
In the expression for y'+hy as a linear combination of x\, . . . , z n +i let us 
replace u' p by the expression given therefor by equation (68). Then let 
us denote the coefficients of Xi, . . . , x n+ i by yi, . . . , y n +i respectively, so 
that 2/1, . . . , y n +i are local coordinates of the point y'+hy on an inter- 
sector tangent referred to the pyramid whose vertices are the points 



57] SURFACES AND VARIETIES 279 

#!,..., n +i. Thus we find the parametric equations of the locus of the inter- 
sector tangents at the points of a fixed generator Sk, 



k+l 



(p = 2, . . . , AJ+!) 
= k+2, . . . , n 



(69) 



If 2k>n 2, homogeneous elimination of w 2 , . . , u k +i and /i from these 
equations gives n 1 k quadratic equation of the locus of the intersector 
tangents, 

ft+i 
(70) 2/ ; (a 7 g2/ n+ i~a / , n +i2/ 9 ) = (q = k+2, . . . , n) . 



If, however, 2k ^ n 2, homogeneous elimination of w ; from the last group 
of equations (69) gives nl 2k linear equations; then k quadratic equa- 
tions can be obtained as before. 

We turn now to the consideration of a special case. The configuration 
composed of two ruled surfaces with their generators in one-to-one corre- 
spondence, and with corresponding generators skew to each other, is of fre- 
quent occurrence in the geometry of surfaces in ordinary space. Such a con- 
figuration can be studied by means of the equations (62) when n = 3 and the 
transformation (65) is used. The differential equations can be reduced by 
this transformation to the canonical* form illustrated by figure 41, 



Let us denote the two surfaces by Ri 2 and R^, their generators being the 
lines XiX% and a^. The intersector curves on the ruled surface Ru with re- 
spect to the ruled surface #34 are those curves such that the tangent line at 
each point x\-\-ux^ on a generator x\x%, of each of them intersects the corre- 
sponding generator 324. The differential equation of the intersector curves 
on the surface Ri 2 is found to be an equation of Riccati, 

(72) u'= au+fan 

* Lane, 1923. 4, p. 284. 



280 



PROJECTIVE DIFFERENTIAL GEOMETRY 



This differential equation becomes u' = when the canonical form (71) is 
being used; hence the curves u~ const, are intersector curves on the surface 
#12; a similar remark can be made for the surface #34. The locus of the 




FIG. 41 

intersector tangents of the surface R^ at the points on a generator 1X2 has 
the local equation 



(73) 



- 0132/12/4+0242/22/3 - 0232/22/4 = 



and is therefore a quadric surface. 

The foregoing discussion can be generalized in the following way. If in 
(61) the coordinates x{ of the ra+1 points # are now supposed to be func- 
tions of m (and not fewer than m) independent variables t l , . . . , t m (m<n), 
the locus of each of the points is a variety V m , and we thus obtain n+1 such 
varieties with their points in correspondence. It is possible to determine 
the coefficients of a system of partial differential equations of the form 



n+l 



(74) 



so that 



(i = l, . . . , n+l; p = l, . . . , m) 



57] SURFACES AND VARIETIES 281 

will be n+ 1 sets of solutions. These coefficients satisfy m(m-l)(n+l) 2 /2 
integrability conditions which can be obtained by demanding that the ex- 
pressions for the derivatives of the second order calculated by differentiating 
equations (74) shall be unique. The integrability conditions are 

n+l n+l 

da tjp /dt*+ Va 



(i,; = l, . . . , n+l; p, g=l, . , m;p<q) , 
and from them it is easy to deduce the equations 

n+l 

(76) ^(daup/dtt-daiit/dP^Q (p, <?=!, . . . , m; p<q) . 
t-i 

By means of a transformation of the form (66) with coefficients that are 
functions of t l , . . . , t m j and a transformation of independent variables of 
the form 

(77) i* = tt*(* 1 > ...,"0 (p = l, ...,m), 

we can study the protective differential geometry of a configuration which 
consists of a certain number of varieties each of which is the locus of < m 
linear spaces in correspondence. 



EXERCISES 

1. The space 05 of the points (10) is the limit of the space S& determined by the 
tangent plane at a point P x of a surface and the tangent plane at a neighboring 
point Q on a curve C through P X) as Q approaches P x along C. 

2. Consider the space $ 6 which joins the space S(2, 0) at a point P x of a surface S 
to a variable point H of the cubic curve in the space 3 of the points x uuu , . . . , x vvv , 
and prove that when H varies on the cubic, the space $ 6 generates a cone which 
is a variety Ff. 

3. Discuss in detail the neighborhood of the fourth order of a point on a surface 
in space S n . 

4. The space S(2, 1) at a point P x of a variety V m in the direction of a curve 
through P x on V m is a space S m +i. The variety F(2, 0) which is the generalized cone 
of Del Pezzo at the point P x of the variety V m has for its vertex the tangent space S m 
of V m at P x , lies in the space 5(2, 0) of V m at P x , and is of order 2 m ^. 



282 PROJECTIVE DIFFERENTIAL GEOMETRY 

5. A hyperplane of tacnodal section at a point P x of a surface in space S 6 is tan- 
gent to the cone of Del Pezzo at P xy and is also tangent to the cone of Del Pezzo at a 
point consecutive to the point P x on a principal curve through P x . 

SEGRE, 1921. 1, p. 201 

6. At a point of a surface the triple of directions defined by a non-singular binary 
cubic differential form adu*+bdv* = Q has for hessian pair the pair of directions given 
by dudv-0. The directions of the hessian pair separate harmonically the directions 
of the first polar pair a8udu 2 -\-b8vdv* = Q of an arbitrary direction 8v/8u with re- 
spect to the triple. This property is characteristic of the hessian pair. 

7. When the hyperplane S(2, 0) at a point consecutive to a point P x on a curve 
in the direction dv/du at P x , on a surface F in space $ 5 , intersects the tangent plane 
of F at P x in the tangent line of a curve in the direction dv/8u through P x , then the 
direction dv/du is one or the other of the polar pair (31) of the direction dv/8u with 
respect to the triple of directions (24) . 

8. Construct a theory of surfaces in space $4 using a system of equations of the 
form 

x uv =cx-\-ax u +bx v , 



Find the conditions of complete integrability, and compute power series expansions 
for a point on the surface referred to the local pyramid of reference with vertices at 
the points x, x u , x v , x uu , x vv . Considering the effect of the transformation 

x = \x , u=U(u) , v=V(v) , 

reduce the system of equations to canonical forms; find a co variant pyramid of refer- 
ence, and a canonical form for the power series expansions. 

9. On a surface of Exercise 8, find the differential equation of the < 2 quasi- 
asymptotic curves each of which is such that at each of its points its osculating 
space 83 coincides with the space S(2, 1) in the direction of the curve at the point. 
Prove that every space $3 containing the tangent plane at a point of the surface is 
the osculating space 83 of two of these quasi-asymptotic curves through the point, 
whose directions separate the conjugate directions at the point harmonically. 

BOMPIANI, 1912. 2, p. 404 

10. Considering those surfaces of space 84 which sustain one-parameter families 
of asymptotic curves, interchange x uv and x uu in the fundamental equations of Exer- 
cise 8 and reconstruct the theory. 

11. Two consecutive spaces S(k, 0) intersect in a space S(k 1, 0) and are con- 
tained in a space $(&-|-l, 0). 



SURFACES AND VARIETIES 283 

12. Extend the correlation of Chasles to read as follows. The tangent spaces 
$b+i at the points on a generator Sk of a variety Vk+i correspond projectively to 
their points of contact. 

13. Discuss the variety generated by the point z given in equation (51) when w 
and dtf/dr vary and r is fixed. This variety is ordinarily of dimensions k+m 1 and 
of order Ck+ m -i,k; it is of class C&+i, m __i or C m ,k according as m^.k+1 or m^k+l. 

SEGRE 1910. 1, p. 93 

14. On a surface x = x(u, v) in space 85 consider three consecutive points P x , PI, P 2 
of a curve belonging to a one-parameter family of curves. The tangent planes of the 
surface at the points P\ t Pi determine a space S* which intersects the tangent plane 
of the surface at the point P x in a straight line. Denoting the direction of this line 
by 5v/du, show that 

(x, x u , x v , x uu du-\-x uv dv, x uv du+x vv dv , 



dy? + 2x uuv dudv + x uvv dv 2 + 

+ 8v[x uuv du 2 +2x uvv dudv+x vvv dv 2 +x uv d?u+x vv d?v])==Q . 

(The oo i given curves and the < 1 curves defined by this equation are said to be in 
the relation of conjugacy of the second kind.) Those curves on the surface which are 
self-conjugate in this sense are the principal curves. BOMPIANI, 1922. 6 

15. Using the usual local coordinate system in the space S(2, 0) at a point x of a 
surface S, show that the equation of the space $4 which is tangent, at the point y 
of equation (55), to the variety V* which is the locus of the tangent planes of S is 



Hence show that the one-parameter family of these spaces $4 at all points of the 
tangent plane at the point x of the surface S envelop the cone of Del Pezzo, 



16. If a surface immersed in space S& is non-developable and is not a surface F, 
and if the variety F 4 which is the locus of its tangent planes admits only oo 2 tangent 
spaces 5 4 , then the surface is a surface of Veronese. SEGRE, 1907. 2, p. 1078 

17. Consider a curve C on a non-developable surface S, which is not a surface F 
and is immersed in space S 6 . Consider the variety F 3 which is the locus of the 
tangent planes of S at the points of C. Consider, further, the space 84 which is 
tangent to the variety Fa along the tangent plane at a point x of the surface S. This 
space S* has four-point contact with the curve C at the point x and consequently 
contains the osculating space S 3 at the point x of the curve C if, and only if, the 
curve C is a principal curve. SEGRE, 1921. 1 

18. A plane curve on a surface immersed in space /S 5 is a principal curve. If a 
non-developable surface immersed in space S$ has on it < 2 plane curves, the prin- 
cipal curves are indeterminate, and the plane curves are conies. 



284 PROJECTIVE DIFFERENTIAL GEOMETRY 

19. Calculate the integrability conditions of system (33), and prove that if the 
principal curves are indeterminate on a non-developable surface immersed in space 
$5, the surface is a surface of Veronese. 

20. At a point x of a surface x=x(u, v) in space S n consider the parametric 
curves C u , C v . The ruled surface R u of w-tangents constructed at the points of the 
curve C v has for tangent space 83 along the generator xx u the space 83 determined 
by the points x, x u , x v , x uv . The ruled surface R v of v-tangents constructed at the 
points of the curve C u has along the generator xx v the same tangent space 83. 

21. There are two lines that are tangent to both of two ruled surfaces Ru, Ru 
with their generators in one-to-one correspondence in space $3, the two points of 
contact with Rn being on any prescribed generator xiX 2 , and the two points of con- 
tact with #34 being on the corresponding generator z 3 z 4 . Determine these four points 
of contact. LANE, 1923. 4, p. 291 

22. When a point x varies on a curve of the family dv \du-0 on an integral 
surface of equations (III, 6), the line joining the point x to the point y defined by 
equation (III, 35) generates a ruled surface R xv , and the reciprocal line joining the 
points p, <r defined by equations (III, 42) describes a ruled surface R pff . These sur- 
faces have their generators in one-to-one correspondence, and the sixteen coefficients 
of the system of equations of the form (62) for these surfaces have the following ex- 
pressions when xi = x, Xz = y, Z S =P, 4=0-: 



23. Using equations (62) and the transformation (65) with n = 5, study a triple 
of ruled surfaces R\i, #34, RM in space & with their generators X\X<L, x^x*, x&* in 
correspondence. In particular, show that on the ruled surface R\z there are two 
curves such that at each point y, defined by y = x\+uxi on a generator x\x^ of each 
of them its osculating plane meets in a straight line the space 83 of the corresponding 
generators XsX*, x&t. These curves are generated by the points y for which u is a 
root of the equation 

3 

A 

= 0. 

LANE, 1928. 5, p. 790 

24. At the conclusion of Section 57 let n = 3, m = 2, and develop a theory of pairs 
of congruences in ordinary space with their generators in one-to-one correspondence. 



SURFACES AND VARIETIES 285 

Calculate the thirty-two coefficients of the system of equations of the form (74) 
for a pair of reciprocal congruences associated with a surface in ordinary space 
(see Ex. 22). COOK, 1930. 2 

25. Consider an integral surface of the system of equations 



x uvv = a'x vu +h'x uv +b'x vv +l'x u +m'x v +d'x , 

and suppose that this surface is not an integral surface of any equation of the 
second order. Two of the eight integrability conditions of this system are a'=6=0. 
Consider, at a point of the integral surface, the two ruled surfaces R u , R v described 
in Exercise 20. Consider, on the surface R u , all the curves that intersect the generator 
xx u , and prove that the osculating planes of these curves at the points of the generator 
xx u lie in a space 84. Prove the symmetric theorem with u and t; interchanged. 

BOMPIANI, 1919. 2, pp. 362-63 

26. The oo 2 spaces $3 defined in Exercise 20 at the points of an integral surface 
of the system of Exercise 25 can be arranged in two ways, along the w-curves and 
along the ^-curves, as the oo 1 osculating spaces 83 of a 1 curves. 

BOMPIANI, 1919. 2, p. 633 

27. As a point x varies along a v -curve on an integral surface S of the system of 
Exercise 25, the point x u b'x describes a curve characterized geometrically by the 
property that its osculating plane at each of its points is in the space $3 tangent, along 
the line xx u through the point, to the ruled surface R u . Asu,v vary, the point x u b'x 
describes a surface of the same type as the surface S. The point x v ax describes 
similarly a transform of the surface S in the ^-direction. 

BOMPIANI, 1919. 2, p. 634; P. TZITZEICA, 1927. 2 

28. In the space *S 6 of the points x UU) . . . , x vvv of Section 52 the parametric equa- 
tions of the varieVy FS, which is the locus of the plane determined by the points x uu + 
x uv \ x uv +x vv \, H, can be written in the form 



Writing three linear equations which represent a fixed space 8 in the space , prove 
analytically that the variety F 3 meets the space 8 in five points, thus proving again 
that the variety F 3 is of order 5. 

29. The space S(2, 0) at a point P x of a surface F is filled by the osculating planes 
at P x of all the curves on F that pass through P x . 



CHAPTER VIII 
MISCELLANEOUS TOPICS 

Introduction. The purpose of this chapter is to include certain topics 
which it does not seem desirable to omit altogether, and which nevertheless 
do not find adequate presentation elsewhere in this volume. Section 58 is 
devoted to some historical remarks sketching the beginning and growth of 
projective differential geometry. The method of differential forms is dis- 
cussed in Section 59 more extensively than hitherto in these pages, par- 
ticular attention being given to the way in which the method has been 
applied by Fubini in the study of surfaces in ordinary space. In Section 60 
certain coordinate systems are compared and the equations of transforma- 
tion between them are found. Finally, in Section 61 a brief account is given 
of Wilczynski's theory of congruences in ordinary space, a subject so inti- 
mately connected with the theory of surfaces that it seems appropriate to 
include it here. 

68. Historical remarks. The purpose of this section is to recapitulate 
briefly the story of the beginning and growth of projective differential 
geometry. There was first of all a period of discovery of isolated theorems 
which were of a projective differential nature but which were not recog- 
nized at the time as having this character, because there was then no organ- 
ized science of projective differential geometry. Later came the initiation 
of projective differential investigations, and still later the organization of 
comprehensive theories and the perfection of systematic methods of study. 

Perhaps the simplest and earliest example of a configuration whose defi- 
nition is of a projective differential nature consists of the tangent line at a 
point of a curve. The definition that we use today does not go back to the 
Greeks, who seem to have thought of the tangent at a point of a curve as 
merely cutting the curve in one point at its point of contact. Our definition 
of the tangent as the limit of a secant probably dates no farther back than 
the works of Fermat (1601-65) and Descartes (1596-1650). 

Since a developable surface is the locus of the tangents of a curve it is 
natural, and quite probable, that the theory of developable surfaces bears 
the next earliest date. Cajori* says: "The first critical studies of develop- 
able surfaces were made by Leonhard Euler and Gaspard Monge. The two 
investigators approached the subject about the same time, but Euler's 

* Cajori, 1929. 3, p. 432. 

286 



58] MISCELLANEOUS TOPICS 287 

paper* received earlier publication, in 1772. It is noteworthy that at this 

time Euler was blind About the same time, and independently of 

Euler, the subject of developable surfaces was investigated by Gaspard 
Monge, the creator of descriptive geometry. His earliest publication on 
such surfaces appeared at Paris in 1785; he discussed them repeatedly in 
later writings. Mongers treatment is less analytical than that of Euler and 
more nearly the result of direct contemplation of space relations." We may 
remark that Euler (1707-83) seems to have been the first to consider a de- 
velopable surface as the locus of a one-parameter family of lines, and that 
Monge (1746-1818) seems to have been the first to consider a developable 
as the envelope of a one-parameter family of planes. 

Monge made still other contributions to projective differential geometry. 
For example, he gavef the first propositions on general congruences of 
straight lines. In particular, he discovered the two focal points on each 
generator of a congruence, and the two families of developables in a con- 
gruence. Moreover, both in the memoir last cited and in his Geometrie 
descriptive (1798) he studied the lines of curvature on a surface which, to 
be sure, are not capable of a purely projective definition, but which are 
closely related to certain aspects of the projective theory. Monge is said 
to have been an inspiring teacher, and it is certain that some of his pupils 
distinguished themselves as geometers, among them being Malus, Dupin, 
Brianchon, Poncelet, and Plucker. 

It seems appropriate to make a few remarks here on the chronology of 
the geometry that has the straight line as generating element. Malus in his 
Optique seems to have originated}: the concept of a general complex of 
straight lines, but the name is due to Plucker. Giorgini was probably the 
first to consider, in 1827, the linear complex, which was studied in 1833 
by Mobius arid in 1837 by Chasles. Pliicker in his System der Geometrie des 
Raums (1846) was the first to take the straight line as the generating element 
of ordinary space, but it was Grassmann who, in his Ausdehnungslehre 
(1844), defined the projective homogeneous coordinates of a linear subspace 
Sk of a linear space S n (k<ri) by means of the coordinates of fc-f-1 inde- 
pendent points in the space &. In the special case n = 3, k = 1 we have the 
line coordinates commonly called pliickerian although Grassmann seems 
to have defined them first. Plucker used quite a different system of line 
coordinates in his early investigations of ordinary ruled space. It is worthy 

* Euler, 1772. 1. 

t Monge, 1781. 1. 

J Journal de Vecole poly technique, Vol. VII (1808). 

Memorie di matematica della societd italiana delle scienze (Modena). 



288 PROJECTIVE DIFFERENTIAL GEOMETRY 

of note that Cayley in 1859 independently discovered Grassmann's coordi- 
nates of a straight line. 

We have already had occasion to refer to various isolated results of a pro- 
jective differential nature. We recall the work of Dupin on conjugate tan- 
gents and asymptotic tangents, bearing the date 1813. The correlation of 
Chasles dates back to 1839, and the theorem of P. Serret on ruled surfaces 
to 1860. Moutard discovered the quadric that bears his name in, or about, 
1863. Hermite's theorem on the contact of a quadric surface with an ana- 
lytic surface was published in 1873. Lie announced the discovery of the 
quadric that bears his name in 1878. And in 1880 Darboux described the 
tangents now called the tangents of Darboux. 

In the presence of all these outcroppings of projective differential geome- 
try it is not surprising to learn that at about the beginning of the last quar- 
ter of the nineteenth century the first attempt was made to construct a 
general theory of this subject. Credit for consciously undertaking the first 
systematic projective differential investigation is due to Halphen (1844-89). 
Reference has already been made to his Paris thesis of 1878 on plane curves, 
and to the memoir of 1880 on curves in ordinary space. These publications 
contain very fundamental and far-reaching results; surprisingly little has 
been added to our knowledge of plane and space curves since the time of 
Halphen. 

The distinguished Italian geometer C. Segre (1863-1924) began his 
geometrical researches at the University of Torino in the early eighties of 
the nineteenth century. His interest in projective differential geometry is 
said to have been stimulated by Wilczynski at the Heidelberg Congress of 
1904. Beginning with a very significant memoir in 1907 Segre made im- 
portant contributions to the subject. He was not only interested in the 
geometry of ordinary space, to which his contributions of the tangents of 
Segre and the cone of Segre have already been studied in this book, but was a 
leader in studying the projective differential geometry of hyperspace. 
Segre gave analytic proofs regularly, but was also an outstanding exponent 
of the synthetic method, making differential properties even in hyperspace 
appear intuitive. This method has been used with great skill and success by 
Bompiani. 

Wilczynski (1876 ) turned his attention to geometry about 1901 and 

by 1906 had established his reputation as a geometer on a firm foundation. 
He contributed a systematic analytic method, namely, the method of differ- 
ential equations, which has been used in a somewhat modified form in the 
greater part of this book. His extensive geometrical contributions have 
already been so thoroughly discussed in these pages that there is no need to 
enlarge on them here. One of the most successful exponents of Wilczynski's 
method was G. M. Green (1891-1919). 



59] MISCELLANEOUS TOPICS 289 

Fubini (1879 ) became interested in projective differential geometry 

about 1914. He undertook to define a surface in ordinary space, except for a 
projective transformation, by means of differential forms. By 1916 he had 
perfected an analytical method in projective differential geometry, namely, 
the method of differential forms. His name is associated with that of one of 
the distinguished followers of his method, Edouard Cech, through their 
collaboration in publishing the well-known treatise.* Fubini's method is 
employed in their treatise, and we shall consider it again in the next section. 

69. The method of Fubini. In the introduction to Chapter III we made 
some general comments on Fubini's method, in connection with the theory 
of surfaces in ordinary space. The purpose of this section is to follow up 
those remarks with a somewhat more detailed and explicit exposition of 
Fubini ; s method as applied to surfaces in ordinary space, and to make some 
further explanations of the power and limitations of the method. 

We shall assume that the reader knows a little of the theory of differential 
forms. The absolute calculus of Ricci is very useful in this theory. In par- 
ticular, the process of covariant differentiation with respect to a fundamen- 
tal quadratic differential form is frequently employed. The formulas are 
very much simplified by use of the summation convention of tensor analy- 
sis, summation being understood with respect to any index that appears 
twice in the same term. 

We propose, with Fubini, to define a surface in ordinary space by means 
of differential forms. Let us consider a fundamental binary quadratic 
differential form G, 

(1) G = a %] du l du j (an = a/; i, j = 1, 2) . 

The coefficients a*/ are functions of the two variables w 1 , u 2 in a certain re- 
gion, and the discriminant A of the form is supposed not to vanish in this 
region. The form G will be more completely specified later on. Let us also 
consider in space $ 3 a surface S whose parametric vector equation hi pro- 
jective homogeneous coordinates is 

(2) x = x(u\ u 2 ) . 

Let us now define two differential forms Ft, <f> 3 by the equations 

(3) F, = (a?, *i, *,, d*x) | A | - 1 / 2 , $3 = (x, xi, X* ffix) \ A \ -v , 

wherein numerical subscripts of x denote covariant differentiation with re- 
spect to the form G, parantheses indicate determinants of the fourth order 

* Fubini and Cech, 1926. 1 and 1927. 1. 



290 PROJECTIVE DIFFERENTIAL GEOMETRY 

of which only a typical row is written in each case, and vertical bars indi- 
cate absolute value. The forms F 2 , $3 are of the first and second orders 
respectively, since F 2 is independent of the second differentials of u l , u?, and 
<i>3 is independent of the third differentials. 

Both forms F 2 , $3 can be shown by direct calculations, which we shall 
omit, to be absolutely invariant under all proper transformations of parame- 
ters, 

(4) u l = u l (u\ u 2 ) , u 2 = u 2 (u\ u?) . 

To express this invariantive property the forms F 2 , $3 are called intrinsic. 
One undesirable feature of the present situation is that the forms F 2 , $ 3 
are of different orders. It is possible to replace <f> 3 by an intrinsic form of the 
first order. In fact, a little calculation, which again we shall omit, suffices 
to show that the form / 3 defined by 

(5) /3 = 2$ 3 -3^ 2 

is actually of the first order, being independent of the second differentials 
of u l , u 2 . We now have two binary differential forms F%, fs one quadratic 
and the other cubic. These forms are independent of the parametric repre- 
sentation of the surface S, but still depend on the form G and on the pro- 
portionality factor X of the homogeneous coordinates x. 

We wish to find precisely how the forms Ft, / 3 , and incidentally <J> 3 , de- 
pend on the form G and the factor X. Replacing G by another form G' and 
indicating the new expressions by accents, we find 

(6) F^RF 2) 

where 

(7) 

Moreover, multiplying each coordinate x by a factor X, i.e., making the 
transformation 

(8) Z = Xz , 

and distinguishing the new expressions by dashes, we find 

(9) F 2 = X 4 F 2 , i 3 



Since the form/ 3 is not transformed in the same way as the form F% under 
these transformations we inquire whether it is possible to replace fa by 



59] MISCELLANEOUS TOPICS 291 

another form having all the desirable properties possessed by/ 3 and having 
also the property of being cogredient to F 2 under these transformations. 
The answer is in the affirmative. Let us denote the discriminant of F 2 by D, 
and define a form F s by the equation 



(10) F 8 = 2$ 8 -3dF2+(3/4)*Vflog (D/A) 



Now it is easy to verify that under the transformations and with the nota- 
tions of the preceding paragraph we have 

(11) F' t **RFi, F^VFi (* = 2,3). 

Thus we reach the conclusion: 

The forms F^ F s are of the first order and of degrees indicated by the sub- 
scripts, are intrinsic, and are cogredient and relatively invariant under change 
of fundamental form G and the transformation (8) of proportionality factor. 

We next choose the proportionality factor X in a special way. Precisely, 
we choose X so that the discriminant of the. form F s shall be a non-zero con- 
stant times the cube of the discriminant of F 2 . That it is possible to make 
this choice is evident when one considers that the discriminant of F 3 is of 
the fourth degree in the coefficients of F 3 while the discriminant of F 2 is 
quadratic in the coefficients of F 2 . Denoting these discriminants by A, D 
respectively we find 

(12) A = X 16 A, D = \*D . 

Then in order to have A = const. S 3 it is sufficient to choose X so that 

(13) t A = const. X 8 D 3 . 

We are supposing ZMO. It follows that A-^0, and in particular that F 3 is 
not the cube of a linear factor. We denote by <p 2 , ^ the two forms F 2) F z given 
by the second of equations (11) with X satisfying (13). 

The fundamental form G has hitherto been arbitrary. We now choose G 
to be the form <p z . With this choice of G, direct calculation, which we shall 
omit, shows that the hessian of the form <p 3 is proportional to <p 2 , so that <?$ 
is apolar to <p 2 . The coordinates x as now normalized are Fubini's normal 
coordinates. The forms <?2, <p 3 are not only intrinsic but are covariant to the 
surface. 

We can simplify the analysis by taking the asymptotic curves on the 
surface S as parametric. The reader may have observed already that the 
curvilinear differential equation of the asymptotic curves is <f> 2 = 0. Let us write 

(14) <p* 



292 PROJECTIVE DIFFERENTIAL GEOMETRY 

It follows that we must have, except possibly for a constant factor, 



(15) ^3 

Therefore the curvilinear differential equation of the curves of Darboux is 
*> 8 =0. 

The question now is whether these two forms are sufficient to determine 
the surface S except for a projective transformation. The answer is in the 
negative. It is not difficult to show that the coordinates x satisfy equations 
of the form (III, 6), but that the surface is now determined only except for a 
projective applicability, the coefficients p, q of (III, 6) being not yet deter- 
mined. // we adjoin to the forms <pz, v$ the form \l/z defined by 

(16) ^ 2 = pdw 2 -^ 2 , 

these three forms suffice to determine the surface S except for a projective trans- 
formation, since now all the coefficients of equations (III, 6) are determined. 
The form ^ 2 vanishes for the curves corresponding to the developables of 
the congruence which is reciprocal to the projective normal congruence of 
the surface S. 

Fubini's method, which we have just explained for surfaces in ordinary- 
space, has also been used by Sannia in studying plane and space curves; by 
Cech in his theory of ruled surfaces; and by Fubini in studying congruences 
and complexes in ordinary space, surfaces in space $ 4 , and hypersurfaces in 
space S n . This method is available for any configuration that has a quad- 
ratic form covariantly connected with it. The method has failed for a 
variety Vk in space S n with l<k<n 1 and n>4, either because of the 
lack of a covariant quadratic form, or because of the lack of an absolute 
calculus for an n-ary p-adic differential form. The method of Wilczynski is 
theoretically available even in these cases but the labor involved in applying 
this method seems to be largely prohibitive. 

60. Comparison of certain coordinate systems. Aside from minor varia- 
tions in notations and conventions which it is easy to recognize, there are 
two or three different local coordinate systems which have been used in 
studying surfaces in ordinary space and which it may be helpful to discuss 
here. We shall first of all compare the local coordinate system used in 
Chapter III with a local coordinate system much used by Fubini and 
Bompiani. Then we shall show how to pass from the notation and local 
coordinate system habitually employed by Wilczynski to that of Chapter 
III. Finally, we shall show how to transform the equations of Wilczynski's 
canonical quadric and cubic from his canonical coordinate system to the 
system used in Chapter III, and shall comment on the results. 



60] MISCELLANEOUS TOPICS 293 

Early in Chapter III we chose as vertices of a local tetrahedron of refer- 
ence at a point x of an integral surface of equations (III, 6) the four co- 
variant points x, x u , x v , x uv , and chose the unit point so that a point X 
defined by 

(17) X = XiX-\-X2X u -\-X3X v -\-X4X uv 

should have local coordinates proportional to xi, . . . , x\. Fubini and Bom- 
piani frequently use* as local coordinates of the same point X the four de- 
terminants 12, Nt, Ni, T defined by 



(18) { 



12 = (X, x uj x v , x uv ) , N 2 =(X, x, x v , x uv ) , 

Ni=(X, x, x u , x uv ) , T=(X, x, x u , x v ) . 



If the expression for X in equation (17) is substituted in equations (18) it is 
easy to see that the equations of transformation between these two systems of 
coordinates are 

(19) pzi = 12, pX2=-Nz, px 3 = Ni, pXt=-T, 

where p is a proportionality factor (see Ex. 39 of Chap. III). These equa- 
tions are quite useful in comparing results of different authors. 

The reader who wishes to compare some of the original memoirs of Wil- 
czynski and his followers with their results as reported here may find the 
following remarks helpful. Wilczynski originally wrote the equations de- 
fining a surface protectively in ordinary space in the form 






the asymptotic curves obviously being parametric. He frequently used a 
canonical form of these equations (see Ex. 2 of Chap. Ill), namely, 






and at the same time employed the local tetrahedron whose vertices are the 

points y, 2/M, 2/ v , yuv Only the first of these points is covariant to the surface. 

It can be shown, f although we shall not reproduce the proof here, that a 

local equation written in Wilczynski's canonical notation and referred to the 

* Fubini, 1918. 1, p. 1035. f Lane, 1926. 10, p. 369. 



294 PROJECTIVE DIFFERENTIAL GEOMETRY 

corresponding local coordinate system (in which a point has coordinates 
yi, . . . , 3/4) can be transformed into the notation of Fubini's canonical 
form and referred to the corresponding local coordinate system (in which 
a point has the coordinates xi, . . . , x* that have been used in this book) 
by proceeding as follows. First replace /, g by the expressions 

(22) /=c-a tt -a 2 -2&6' , g = c'-bi-b'*-2aa' . 

Then make the transformation 

(23) 

Finally make the substitution 

/a b c a' b' c'\ 

(24) \-O u /2 -ft/2 -p -7/2 -6 v /2 -q) ' 

The inverse transformation is not difficult to formulate and will be left to the 
reader. 

At the end of Section 18 reference was made to Wilczynski's definition of 
his canonical quadric, which we shall now explain. Wilczynski used* in 
place of our expansion (III, 18) an expansion 

(25) Z = a^+(s+p)/6+(/#+Jp 4 )/24+ 

where /, J are two absolute invariants which we do not need to define any 
more explicitly at the moment. The coordinate system is covariant. With- 
out giving here the complete geometrical description of the coordinate sys- 
tem we note that the three edges through the point P(0, 0, 0) under con- 
sideration are the asymptotic tangents and the directrix di, while the direc- 
trix d z in the tangent plane is still another edge. It was in completing the 
geometric description of the coordinate system that Wilczynski was led to 
introduce his canonical quadric and cubic. It can be shown, but we shall 
not give the details of the argument, that the transformation from this 
coordinate system to the coordinate system used in this book is capable of 
being decomposed into the following three steps, when the differential 
equations are in Fubini's canonical form : 

y=-2y', z =4*' , 

y' = (Y+BZ)/D , 



(26) 

* Wilczynski, 1908. 2, p. 103. 



60] MISCELLANEOUS TOPICS 295 

where 

fD~l+BX+AY+(AB+e w /2ft)Z , 



Wilczynski's definition of his canonical quadric depends on his canonical 
cubic, whose equation in his coordinate system is 

(28) * = TB+(#+p)/t. 

By the transformation (26) this equation becomes, in our homogeneous 
coordinates, 

f 



This cubic was defined by Wilczynski by means of the following properties. 
It has a unode, i.e., a point where the tangent plane is indeterminate and 
the polar quadric cone is two coincident planes, at the point (0, 0, 0, 1) on 
the directrix d\ through the point P, such that the uniplane, zi = 0, contains 
the reciprocal directrix d z . It has third-order contact with the surface at the 
point P, and the four tangents of fourth-order contact form a harmonic set 
in which conjugate pairs are actually conjugate tangents of the surface. 
The reader may refer to the Second Memoir of Wilczynski just cited for the 
proofs of these properties. 

In Wilczynski's coordinate system the equation of his canonical quadric is 

(30) , 2-^ = 0. 

By the transformation (26) this equation becomes 

(31) 



The essential part of Wilczynski's definition of this quadric can be formu- 
lated by saying that this quadric is the only quadric of Darboux that is tan- 
gent to the uniplane, 

(32) 2xi+vxt+tx*+(e w +rt/2)Xi = Q , 

of the canonical cubic. This definition used by Wilczynski should be com- 
pared with the very elegant definition given nearly two decades later by 
Bompiani, as stated in Section 18. 



296 PROJECTIVE DIFFERENTIAL GEOMETRY 

Incidentally, we may note that the fixed uniplane has a variable pole with 
respect to a variable quadric of Darboux at a point P, and that the locus 
of this pole is the directrix d\ through P. When the pole is on the plane, 
the plane is tangent to the canonical quadric at the point distinct from P 
where the directrix di pierces it. 

The process used by Wilczynski in truncating his canonical expansion to 
obtain a canonical quadric and a canonical cubic suggests the following de- 
velopments. Let us consider our canonical expansion (III, 18), for which 
the geometric significance of the coordinate system is already known, and 
let us truncate this series after the terms of degree n, obtaining thus a 
canonical algebraic surface of degree n. This surface can be characterized 
geometrically by two properties. * First, the algebraic surface has contact of 
order n with the analytic surface (III, 18) at the point P (1,0, 0, 0). Second, 
the algebraic surface has a unode of order n 1 at the point (0, 0, ,1), the uni- 
plane being the plane Xi = 0; this means that the tangent plane and the polar 
surfaces of the algebraic surface at (0, 0, 0, 1) up to and including the one 
of order n 2 are indeterminate, while the one of order n 1 degenerates 
into the plane i = counted n 1 times. 

61. Congruences in ordinary space. The purpose of this section is to 
outline the theory of rectilinear congruences in ordinary space as developed 
by Wilczynski in his prize memoir, f We shall preserve Wilczynski's nota- 
tions as far as feasible, and shall present some of the fundamental geometri- 
cal results. 

The defining system of differential equations can be established in the 
following way. It was remarked in Section 32 that a congruence in space S n 
can be represented analytically by a pair of equations of the form 



With this representation the surfaces S y , S, are the focal surfaces (supposed 
distinct) of the congruence; these surfaces are referred to the conjugate 
nets dudv Q in which the developables of the congruence touch them; a 
generator yz of the congruence touches S v , S z in two points P v , P z with the 
same curvilinear coordinates u, v; and the proportionality factors of y, z 
have been chosen in a special way. When the congruence is in ordinary 
space 3 two more equations appear. The points y, z, y u , z v are not ordi- 
narily coplanar, since two distinct surfaces do not ordinarily have the same 
tangent planes. Then the determinant (y, z, y u , z v ) does not vanish, and it 

* Lane, 1927. 10, p. 812. 
t Wilczynski, 1911.3. 



61] MISCELLANEOUS TOPICS 297 

follows that y uv and z vv can be expressed as linear combinations of y, z, y u , z v . 
The result can be stated as follows. 

A congruence in ordinary space is an integral congruence of a system of 
equations which can be reduced to the form 



(33) 



= a'y+b'z+c'y u +d'z v 



in which the coefficients are scalar functions of u, v. 

It will be left to the reader to show that any projective transform of the 
congruence yz is also an integral congruence of the system (33), and that 
the coefficients of this system satisfy eight integrability conditions obtain- 
able from the equations 



Two of these conditions imply c v = d' u . Consequently there exists a function 
/ defined, except for an additive constant, by the differential equations 

(34) / tt = c, f v = d'. 

The seven remaining independent integrability conditions can now be written 
in the form 



b=-d v -df v , a'=-c' u -c'f u , 
m uu +d vv +dS V v+d v f v fumu = ma+db r , 
n vv +c' uu +c'f uu +c' u f u f v n v =c'a+nb' , 



(35) 



The most general transformation leaving the form of system (33) in- 
variant is 

(36) 2/ = X!/, z = M2> u = a, t5 = (XMa tt ft,^0) , 

where X, a are functions of u alone, and M> ft of v alone. The effect of this 
transformation on system (33) is to produce another system of the same 
form whose coefficients, indicated by dashes, are given by the following 
formulas: 



m = w/i/Xft, , n 

5- (l/ol)(a+<^A- WX) , 6' 

5 = (M/Xal) (6+cW/O , a' = (X//X/3J) (a'+cX/X) , 



298 PROJECTIVE DIFFERENTIAL GEOMETRY 



(37) 



It is possible to find a pair of equations of the form described in Exercise 
13 of Chapter IV for the surface S v , and similarly for S g . In fact, elimination 
of z from system (33) and some of the equations obtainable therefrom by 
differentiation leads at once to the equations for S v , 



/QO\ I V" & ' ^ o ^Vtt2/V > 

1 " ' " '" ; ).W+<Wm 



The equations for the surface S, can be written, whenever they are needed, 
by means of the following substitution: 



u * b c d\ 
v n V a' d' c' ) ' 



Of course, in dividing by m in equations (38) we assume m^O. The first 
equation of system (33) shows that if S y is a proper surface, then m^Q, and 
similarly the second of (33) shows that if S z is a proper surface, then n^O. 
Direct calculation shows that the differential equations of the asymptotic 
curves on S v , S z are respectively 

(40) ddu*+mdv 2 = , ndtf+c'dv* = . 



It follows that if S v is not a developable surface then rf=^0, and if S g is not 
developabk then c'^0. We shall assume hereinafter that mnc'dp^O. 

Equations (40) show that the asymptotic curves on S v and S z correspond 
if, and only if, 



Let us call the left member of this equation the Weingarten invariant W, 
so that we have 

(41) F = ran-c'd=/ ttt , . 



61] 



MISCELLANEOUS TOPICS 



299 



Then we have the theorem : 

The congruence yzisaW congruence if, and only if, f uv = 
Let us define p, a by placing 



(42) 



P = 2/u (log m) u y , 



z v (log n) v z 



Then p is the first Laplace transform of y in the w-direction, and cr is that of 
z in the ^-direction. A system of equations of the form (33) for the con- 
gruence of lines py can be calculated. For this purpose let us place 



Calculating the various derivatives of Y, Z, and eliminating y, z and their 
derivatives, we obtain the desired system of equations, whose coefficients, 
denoted by capital letters, are given by the following formulas: 

M = m[mn (log m) uv ] , N = l/m , 
A=a+(f-2 log m) tttt +(log d/m)(log m-/) tt , 

B = m[a u + bn + af u + dn v (log m) ] 
+ (log m) v [b u +bf u +mnd-b(\og m) u ] 
+w M [/ tttt +/J-/ M (log m) tt -2(log m) utt ] 
(43) <{ -f- (/-flog d/m) u [m uu f v m u am 6(log m) v ] , 

C = (/+log d/m) u , D r = (/+log d/m) v , 

D = d[c'd-(\ogd) uv ], C'=l/d, 



m) v ]/d . 



B'= -(log 



We do not need to write here the corresponding formulas for the congruence 
z<r, since they can be written at once by aid of the substitution (39). The 
congruences yp and za are called the first and minus-first Laplace transforms of 
the congruence yz. 

In order to study the differential geometry of a congruence in the neigh- 
borhood of one of its generators yz it is convenient to introduce line co- 
ordinates. Let us use the tetrahedron y, z, p, a as a local tetrahedron of 
reference with the convention that a point xiy+x^z+Xsp+x^ shall have 
local coordinates proportional to x\, . . . , x^. Then for the local coordinates 
y ly . . . , t/4 of a point Y near the point P v on the surface S v we find by the 
usual method the following power series expansions: 



300 PROJECTIVE DIFFERENTIAL GEOMETRY 

i = 1 + (log m) M Aw+ [a+/ w (log m) w ]d 



...,_.., . Xlog mn),At; 2 /2+ 

(44) 



The corresponding expansions for the coordinates of the point Z near the 
point P g on the surface S g can be found in the same way, or else can be 
written by means of the substitution (39) augmented by 

<> (2 ' :) 

These expansions are 

= nAw+n(log wn) tt Au 2 /2+Wt>AMAt>-Ha'+c'(log m) M ]At; 2 /2+ , 

(46) i - 3 = nAw 2 /2+c'At> 2 /2+ , 



The local coordinates w lk of the generator FZ near the generator yz of the 
congruence are defined by the familiar formulas (I, 42) and are found to 
be represented by the series 



+[mn+(log m) M (log ri) 9 ]&u&v+[b'+f v (log n) v ]Ay 2 /2+ 
(47) 



= Aw / M Aw 2 /2 (log nJv 



co 23 



We observe that the complex a>i 3 = is special and consists of all lines 
intersecting the line z<r. Similarly, the complex w 4 2=0 consists of all lines 
intersecting the line yp] and the complex w 34 = consists of all lines inter- 
secting the generator yz. The lines common to all three of these complexes 
form two flat pencils, one with its center at the point P v and lying in the 
tangent plane of the surface & at the point P z , and the other with its center 
at P t and lying in the tangent plane of S v at P tf . The lines of these two pencils 
are calkd the central rays of the generator yz. 



61] 



MISCELLANEOUS TOPICS 



301 



The central rays (see Fig. 42) can also be characterized in another way. 
If we write the equation (1, 44) of a general linear complex and demand that 
it be satisfied by the series (47) for c^ identically in Aw, At; as far as the 




FIG. 42 

terms of the first degree, we obtain the conditions 034=^23=014=0. Thus 
the equation of the most general linear complex having first-order contact 
with the congruence yz along a generator l yz is found to be 

Such a complex contains the generator l ys and a consecutive generator of 
every ruled surface in the congruence and containing ly t . There are obvi- 
ously >* such linear complexes; all of them have in common the central 
rays of the generator l vg and no other lines. 

If we go on and seek to determine a linear complex having second-order 
contact with the congruence yz along a generator l vi , we find that such a 
complex exists if, and only if, the congruence is a W congruence. In this 
case its equation is 

(49) d<. 



302 PROJECTIVE DIFFERENTIAL GEOMETRY 

and it is called the osculating linear complex along the generator l yz of the W 
congruence yz. 

There is a unique linear complex associated* with one focal point P y of a 
generator l yz of a congruence yz in the following manner. The complex con- 
tains l vz and all its central rays through P v . Moreover, for every ruled sur- 
face of the congruence through l yz , the complex contains also the generator 
YZ consecutive to l yz and all its central rays through its focal point Y con- 
secutive to P tf . This complex is called the associated linear complex of the 
focal point P v of the generator l yz of the congruence. There is, of course, 
a complex similarly associated with the other focal point P*. In order to 
find the equation of the associated linear complex of the point P y we proceed 
as follows. A linear complex containing the line l yz (1, 0, 0, 0, 0, 0) must have 
au = 0. A central ray through the point P y (1, 0, 0, 0) and an arbitrary point 
(0, h, 0, k) on the line za has coordinates (h, 0, A;, 0, 0, 0), and belongs to the 
complex in case also a 2 3 = 0. The series (47) show that the complex contains 
an arbitrary generator YZ consecutive to l yz in case also ai4=0. The equa- 
tion of the complex now has the form (48). We proceed to calculate the 
coordinates of an arbitrary central ray of a neighboring generator YZ 
through the point F. The local coordinates of the point Z are given by 
equations (46). The local coordinates of the corresponding point S near 
the point a are found in the usual way to be given by 



(50) 



The local coordinates of a point hZ+k2 can easily be written, and then the 
needed line coordinates of the central ray joining the point Y to the point 
are found to be 



(ri = [a'+c'(log m) M ]At>-f , 

0-2 = [mn- (log n) ttr ]Att+[&'+./;(log ri) v -n vv /ri\kv+ 



- log tt) v At>+ 



(*, Jkc'AtH ---- , *, *, -fcwAH ---- , /cAwH ---- ) , 

the unwritten coordinates being not needed for our purpose. These coordi- 
nates satisfy equation (48) identically in h, k and to terms of the first degree 
in Aw, At; in case 



* Waelsch, Zur Infinitesimalgeometrie der Strahlencongruenzen und Fldchen, "Sitzungs- 
berichte der Wiener Akademie der Wissenschaften," C, Abt. Ha (1891), 167. 



61] MISCELLANEOUS TOPICS 303 

Thus the local equation of the associated complex of the point P v of the gener- 
ator yz is found to be 



(51) 

Similarly, the equation of the associated complex ofP z is 

(52) 



These complexes are the same and are the osculating linear complex if, and 
only if, the congruence is a W congruence. 

If the congruence is not a W congruence the associated complexes (51), 
(52) are distinct and determine a pencil of linear complexes, whose equation 
can be written in the form 



The special complexes of this pencil are given by 



and consequently their equations are co 42 = and coi 3 = respectively. The 
first of these, as we have already observed, consists of all lines intersecting 
the line yp, and the second bears the same relation to the line zv. Therefore 
the two lines yp, zv are the directrices of the congruence of intersection of the 
associated complexes (51), (52). These two lines are sometimes called the 
principal focal rays of the generator yz, all of the focal rays being the 
tangents of the surface S v at the point P v and the tangents of S, at P x . 

EXERCISES 

1. Use equations (I, 45) to prove that the planes corresponding to any point of 
space in the null systems of four linear complexes of a pencil themselves belong to an 
axial pencil. Prove that the cross ratio of the four planes is independent of the point 
used, so that it may properly be called the cross ratio of the four complexes. Prove 
that the cross ratio of the complexes (51), (52) and 0*42 =0, wia=0 is mn/c'd. 

2. Calculate the system of four differential equations expressing the four third 
derivatives of the line coordinates w of a generator of a congruence, not a W con- 
gruence, as linear homogeneous combinations of w, , v , , <>, w Discuss the 
situation for a W congruence. WILCZYNSKI, 1911. 3, p. 30 

3. If a congruence belongs to a linear complex, and if the first Laplace trans- 
formed congruence also belongs to a linear complex, then all the Laplace trans- 



304 PROJECTIVE DIFFERENTIAL GEOMETRY 

formed congruences belong to linear complexes. The second, fourth, sixth, etc. 
transformed congruences are protectively equivalent to the original congruence; 
the third, fifth, etc. are protectively equivalent to the first Laplace transformed con- 
gruence. WILCZYNSKI, 1911. 3, p. 69 

4. The following formulas give invariants of the parametric conjugate nets on 
the surfaces S y (left column) and S 9 (right column) of the last section; 

Hymn t H x =mn (log n) uv , 

Ky-mn (log m) uv , K e -mn , 

Wy = - (/+log d/m) uv , W^ = -/ , 

W? = -/ , W ( , v) = - (/+log c'/n) uv , 

>v=a+/ tt (log m) u -m uu /m , ,= -(n/cO&'+AOog n) v -n vv /n\ , 



i= - (2/+log ^m) v , 8i= - (2/-log c'n 3 ) v , 

i= ~(2/-log dm s ) w , 8(R= -(2/+log c' 3 n) M . 



BIBLIOGRAPHY 

This Bibliography contains only books and memoirs that are referred to in the 
text. The reader may consult also the following bibliographies: 

1. A bibliography of Green's publications complete except for one paper (Green, 
1920. 2) : 

WILCZYNSKI. "In Memory of Gabriel Marcus Green/ 7 Bulletin of the American 
Mathematical Society, XXVI (1919), 1. 

2. A bibliography of Segre's publications: 

LOEIA. "L'opera di Corrado Segre," Annali di matematica, ser. 4, II (1925), 1. 

3. Two incomplete bibliographies, one of the works of Bompiani on the geometry 
of ordinary space, and one of the works of Bompiani, Segre, and Terracini on the 
geometry of hyperspace: 

FUBINI and CECH. Geometria proiettiva differenziale, Vol. II. Bologna: Zanichelli, 
1927. (See Appendixes II and III.) 

4. A bibliography containing numerous references to papers published up to 
1923: 

BERWALD. "Differentialinvarianten in der Geometric: Riemannsche Mannig- 
faltigkeiten und ihre Verallgemeinerungen," Encyklopddie der mathematischen 
Wissenschaften, III D, 11. (See Part V, p. 104.) 

5. A bibliography prepared by Miss Sperry of the publications of American, 
Canadian, and Asiatic projective differential geometers, not yet published when 
these lines were written, but to appear soon from the University of California Press. 

6. An extensive bibliography of projective differential geometry in a volume in 
French on this subject by Fubini and Cech, not yet published when this was written, 
but to appear soon from the press of Gauthier-Villars, Paris. 

ABBREVIATIONS 

The following list of abbreviations, which will be used hereinafter, is extracted 
from the list on pages 13-17 of Selected Topics in Algebraic Geometry, Bulletin 63 
of the National Research Council. 

1. Am. /.: American Journal of Mathematics 

2. Am. M.S. Bull.: American Mathematical Society Bulletin 

3. Am. M.S. Trans.: American Mathematical Society Transactions 

4. Ann. di mat.: Annali di matematica pura ed applicata 

5. Ann. of Math.: Annals of Mathematics 

6. Berlin Sitzungsber.: Sitzungsberichte der Berliner Mathematischen Gesellschaft 

7. Boll. d.U.M. It.: Bollettino della Unione Matematica Italiana 

8. Bologna Mem.: Bologna. Reale Accademia delle Scienze dell'Istituto Memorie 

9. Brux. Bull: Bruxelles. Academic Royale de Belgique. Bulletins 

305 



306 PROJECTIVE DIFFERENTIAL GEOMETRY 

10. Butt. sci. m.: Bulletin des sciences mathematiques 

11. C.R.: Comptes Rendus. Academic des Sciences de PInstitut de France. Paris 

12. /. de math.: Journal de mathematiques pures et appliqutes. Liouville 

13. J. EC. Polyt.: Journal de VEcole Poly technique. Paris 

14. Lomb. 1st. Rend.: Reale Istituto Lombardo di Scienze e Lettere. Milano. 
Rendiconti 

15. London M.S. Proc.: London Mathematical Society Proceedings 

16. Math. Ann.: Mathematische Annalen 

17. Math. Zschr.: Mathematische Zeitschrift. Berlin 

18. Nap. R.A. Rend.: Napoli. Reale Accademia delle Scienze Fisiche et Mate- 
ma tiche. Rendiconti 

19. Pal. Circ. Mat.: Palermo. Circolo Matematico. Rendiconti 

20. Rom. Ace. L. Rend.: Roma. Reale Accademia dei Lincei. Rendiconti 

21. TdhokuMJ.: Tdhoku Mathematical Journal 

22. Torino Atti: Torino. Reale Accademia delle Scienze. Atti 

23. Torino Mem.: Torino. Reale Accademia delle Scienze. Memorie 

24. Wash. Nat. Ac. Sc. Proc.: Washington. National Academy of Sciences Proceed- 
ings 

1772. 1. EULEB. "De solidis qvorvm svperficiem in planvm explicare vicet," 
Novi commentarii academiae scientarvm imperatis Petropolitanae pro anno 
1771 (Petropoli), XVI, 3. 

1781. 1. MONGE. "Me'frnoire sur la the'orie des de'blais et des remblais," Histoire de 
VAcademie Royale des Sciences. Annee 1781. Avec les memoires de mathe- 
matique et de physique, pour la m$me annee (Paris, 1784), p. 666. 

1813. 1. DUPIN. Dtveloppements de geometric. Paris: Courcier. 

1839. 1. CHASLES. "Me*moire sur les surfaces engendre*es par une ligne droite: 
particulierement sur rhyperboloide, le paraboloi'de et le c6ne du second 
degre*," Correspondance mathematique et physique, ser. 3, III (old ser., 
XI), 49. 

1860. 1. SERBET, P. Theorie nouvelle geometrique et mecanique des lignes a double 
courbure. Paris: Mallet-Bachelier. 

1870. 1. CHASLES. Rapport sur les progres de la geometric. Publication faite sous 
les auspices du minist&re de Instruction publique, Paris. 

1872. 1. RIBAUCOUB. "Sur la th^orie des lignes de courbure," C.R., LXXIV, 

1489. 

2. KLEIN. "Ueber Liniengeometrie und metrische Geometric," Math. Ann. 
V, 257. 

1873. 1. HEBMITE. Cows d' analyse de VEcole Poly technique, Vol. I. Paris: Gau- 

thier-Villars. 

1879. 1. SALMON. Higher Plane Curves. 3d ed. 

1880. 1. HALPHEN. "Sur les invariants diffe*rentiels des courbes gauches," J. EC. 

Polyt., XXVIII, 1. 

2. DABBOUX. "Sur le contact des courbes et des surfaces," Bull. sci. m. t 
ser. 2, IVi, 348. 



BIBLIOGRAPHY 307 

3. MOUTARD. "Sur le contact des coniques et des surfaces," C.R., XCI, 
1055. 

1885. 1. SBGBE. "Sulla geometria della retta e delle sue serie quadratiche," 

Torino Mem., ser. 2, XXXVI, 87. 

1886. 1. DEL PEZZO. "Sugli spazi tangent! ad una superficie o ad una varieta im- 

mersa in uno spazio di piu dimensioni," Nap. R.A. Rend., XXV, 176. 
2. LEVY, L. "Sur quelques Equations line"ares aux de'rive'es partielles du 
second ordre," /. EC. Polyt., LVI, 63. 

1887. 1. DARBOUX. Lemons sur la theorie generate des surfaces, Vol. I. Paris: 

Gauthier-Villars. 

1889. 1. DARBOUX. Legons sur la theorie generate des surfaces, Vol. II. Paris: 
Gauthier-Villars. 

1892. 1. KOENIGS. "Sur les re*seaux plans a invariants e*gaux et les lignes asymp- 

totiques," C.R., CXIV, 55. 

1893. 1. LAPLACE. (Euvres, Vol. IX Paris: Gauthier-Villars. 

1896. 1. GOURSAT. "Sur les Equations Iin6ares et la me'thode de Laplace," Am. J., 

XVIII, 347. 

1897. 1. SEGRE. "Su alcuni punti singolari delle curve algebriche, et sulla linea 

parabolica di una superficie," Rom. Ace. L. Rend., ser. 5, Viz, 168. 
2. GUICHARD. "Sur les systemes orthogonaux et les systemes cycliques," 
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1904. 1. GOURSAT-HEDRICK. Mathematical Analysis, Vol. I. Boston: Ginn & Co. 

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Trans., VI, 99. 

1906. 1. WILCZYNSKI. Projective Differential Geometry of Curves and Ruled Sur- 

faces. Leipzig: Teubner. 

1907. 1. WILCZYNSKI. "Projective Differential Geometry of Curved Surfaces 

(First Memoir)," Am. M.S. Trans., VIII, 233. 

2. SEGRE. "Su una classe di superficie degl'iperspazi legata colle equazioni 
linear! alle derivate parziali di 2 ordine," Torino Atti, XLII, 1047. 

1908. 1. SEGRE. "Complement! alia teoria delle tangenti coniugate di una super- 

ficie," Rom. Ace. L. Rend., ser. 5, XVII 2 , 405. 

2. WILCZYNSKI. "Projective Differential Geometry of Curved Surfaces 
(Second Memoir)," Am. M.S. Trans., IX, 79. 

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Circ. Mat., XXX 2 , 87. 

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308 PROJECTIVE DIFFERENTIAL GEOMETRY 

3. WILCZYNSKI. Sur la tMorie generale des congruences, "Me*moires publics 
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Vol. III. 

1912. 1. BoMPiANi.-"Sull'equazione di Laplace," Pal. Circ. Mat., XXXIV, 383. 
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Torino Atti, XLVIII, 393. 

1913. 1. SEGRB. "Sulle congruenze rettilinee W di cui una od ambe le falde focali 

sono rigate," Torino Atti, XLIX, 291. 

2. WILCZYNSKI. "On a Certain Class of Self-Pro jective Surfaces," Am. M.S. 
Trans., XIV, 421. 

3. STOUFFBR. "In variants of Linear Differential Equations, with Applica- 
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ser. 2, XI, 185. 

1914. 1. SNYDER and SISAM. Analytic Geometry of Space. New York: Henry Holt 

&Co. 

2. BOMPIANI. "Alcune proprieta proiettivo-differenziali dei sistemi di retti 
negli iperspazi," Pal. Circ. Mat., XXXVII, 305. 

3. WILCZYNSKI. "Uber Flachen mit unbestimmten Directrixcurven," 
Math. Ann., LXXVI, 129. 

1915. 1. WILCZYNSKI. "The General Theory of Congruences," Am. M.S. Trans., 

XVI, 311. 

2. SULLIVAN. "Scroll Directrix Curves," Am. M.S. Trans., XVI, 199. 

3. GREEN. "Pro jective Differential Geometry of One-Parameter Families of 
Space Curves, and Conjugate Nets on a Curved Surface (First Memoir)," 
Am. J., XXXVII, 215. 

4. JONAS. "Tiber die Transformation der konjugierten Systeme und (iber 
den gemeinsamen Ursprung der Bianchischen Permutabilitatstheoreme," 
Berlin Sitzungsber., XIV, 96. 

1916. 1. GREEN. "Projective Differential Geometry of One-Parameter Families 

of Space Curves, and Conjugate Nets on a Curved Surface (Second 
Memoir)," Am. J., XXXVIII, 287. 

2. COOLIDGE. A Treatise on the Circle and the Sphere. Oxford: Clarendon 
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3. GREEN. Abstract, Am. M.S. Bull, XXII, 274. 

4. GREEN. Abstract, Am. M.S. Bull, XXIII, 73. 

1917. 1. GOURSAT-HEDRICK. Mathematical Analysis, Vol. II, Part II. Boston: 

Ginn & Co. 

2. GREEN. "On the General Theory of Curved Surfaces and Rectilinear 
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3. EISENHART. "Transformations T of Conjugate Systems of Curves on a 
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1918. 1. FUBINI. "Fondamenti della geometria proiettivo-differenziale di una 

superficie," Torini Atti, LIII, 1032. 

2. SPERRY. "Properties of a Certain Project! vely Defined Two-Parameter 
Family of Curves on a General Surface," Am. /., XL, 213. 



BIBLIOGRAPHY 309 

3. GREEN. "Plane Nets with Equal Invariants," Ann. of Math., XIX, 246. 

4. HALPHEN. (Euvres, Vol. II. Paris: Gauthier-ViUars. 

1919. 1. GREEN. "Memoir on the General Theory of Surfaces and Rectilinear 

Congruences," Am. M.S. Trans., XX, 79. 

2. BOMPIANI. "Determinazione delle superficie integrali d'un sistema di 
equazioni a derivate parziali lineari ed omogenee," Lomb. 1st. Rend., LII, 
610. 

1920. 1. WILCZYNSKI. "Geometrical Significance of Isothermal Conjugacy," 

Am. J., XLII, 211. 

2. GREEN. "Nets of Plane Curves," Am. M.S. Trans., XXI, 207. 

3. WILCZYNSKI. "One-Parameter Families and Nets of Ruled Surfaces and 
a New Theory of Congruences," Am. M.S. Trans., XXI, 157. 

1921. 1. SEGRE. "Le linee principali di una superficie di S 6 e una proprieta carat- 

teristica della superficie di Veronese," Rom. Ace. L. Rend., ser. 5, XXXi, 
200 and 227. 

1922. 1. LIE. Gesammelte Abhandlungen, Vol. III. Leipzig: Teubner. 

2. CECH. "L'intorno di un punto d'una superficie considerate dal punto di 
vista proiettiva," Ann. di Mat., ser% 3, XXXI, 191. 

3. LANE. "A General Theory of Conjugate Nets," Am. M.S. Trans., XXIII, 
283. 

4. WILCZYNSKI. "Some Generalizations of Geodesies," Am. M.S. Trans., 
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5. BIANCHI. Lezioni di geometria differenziale (3d ed.), Vol. I. Pisa: Spoerri. 

6. BOMPIANI. "Sistemi coniugati sulle superficie degli iperspazi," Pal. Circ. 
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7. CECH. "Sur la ge"ome*trie d'une surface et sur le facteur arbitraire des 
coordone*es homogenes," Rom. Ace. L. Rend., ser. 5, XXXI2, 475. 

8. CECH. "Sur la correspondance ge'ne'rale de deux surfaces," Bulletin inter- 
national, classe des sciences mathtmatiques naturelles et de la m6decine, 
Acadtmie Tcheque des Sciences (Ceskd Academic vd a umeni). Prague. 

1923. 1. BOMPIANI. "Proprieta generali della rappresentazione puntuale fra due 

superficie," Ann. di mat., ser. 4, 1, 259. 

2. BERTINI. Geometria proiettiva degli iperspazi (2ded.). Messina: Giuseppe 
Principato. 

3. EISENHART. Transformations of Surfaces. Princeton University Press. 

4. LANE. "Ruled Surfaces with Generators in One-to-One Correspondence," 
Am. M.S. Trans., XXV, 281. 

5. CARPENTER. "Flecnodal Properties of Ruled Surfaces," Tdhoku M.J., 
XXIII, 104. 

1924. 1. BOMPIANI. "Determinazioni proiettivo-differenziali relative ad una 

superficie dello spazio ordinario," Torino Atti, LIX, 409. 

2. BOMPIANI. "Contributo alia geometria proiettivo-differenziale di una 
superficie," Boll. d.U.M. It., Ill, 49. 

3. TZITZEICA. Geomttrie diff&rentiette protective des reseaux. Paris: Gauthier- 
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310 PROJECTIVE DIFFERENTIAL GEOMETRY 

1924. 4. BOMPIANI. "Sistemi coniugati e sistemi assiali di linee sopra una super- 

ficie dello spazio ordinario," Boll. d.U.M. It., Ill, 10. 

1925. 1. THOMSEN. "Ueber eine liniengeometrische Behandlungsweise der pro- 

jectiven Flachentheorie und die projective Geometrie der Systeme von 
Flachen zweiter Ordnung," Abhandlung aus dem mathematischen Seminar 
der Hamburgischen Universitdt, IV, 232. 

1926. 1. FUBINI and CECH. Geometria proiettiva differenziale, Vol. I. Bologna: 

Zanichelli. 

2. BOMPIANI. "Ancora sulla geometria delle superficie considerate nello 
spazio rigato," Rom. Ace. L. Rend., ser. 6, IV2, 262. 

3. LANE. "The Correspondence between the Tangent Plane of a Surface 
and Its Point of Contact," Am. J., XLVIII, 204. 

4. LANE. "Bundles and Pencils of Nets on a Surface," Am. M.S. Trans., 
XXVIII, 149. 

5. GRAUSTEIN. "An Invariant of a General Transformation of Surfaces," 
Am. M.S. Bull, XXXII, 357. 

6. SANNIA. "Nuova trattazione della geometria proiettivo-differenziale 
delle curve sghembe (memoria 2*)," Ann. di mat., ser. 4, III, 1. 

7. BOMPIANI. "La geometria delle superficie considerate nello spazio rigato," 
Rom. Ace. L. Rend., ser. 6, IIIi, 395. 

8. BOMPIANI. "Invariant! proiettivi di contatto fra curve piane," Rom. Ace. 
L. Rend., ser. 6, IIIi, 118. 

9. BOMPIANI. "Sul contatto di due curve sghembe," Bologna Mem., ser. 8, 
III, 35. 

10. LANE. "Wilczynski's and Fubini's Canonical Systems of Differential 
Equations," Am. M.S. Bull., XXXII, 365. 

11. KLOBOUCEK. "Quadrique osculatrice le long d'une ge*ne"ratrice de la sur- 
face re"gle*e donne*e par trois courbes infiniment voisines. Quadrique de 
Lie. L'element de surface du 3 e et 4 e ordre," Bulletin international, classe 
des sciences maMmatiques naturelles et de la medecine, Academie Tcheque 
des Sciences (Ceskd Academie v&d a umgnt). Prague. 

1927. 1. FUBINI and CECH. Geometria proiettiva differenziale, Vol. II. Bologna: 

Zanichelli. 

2. P. TZITZEICA. "Sur un certain systeme d'e"quations aux de*riv6es par- 
tielles," C.R., CLXXXIV, 582. 

3. LANE. "Quadriche aventi per generatrici le tangenti asintotiche in un 
punto di una superficie," Rom. Ace. L. Rend., ser. 6, Vi, 100. 

4. LANE. "The Contact of a Cubic Surface with an Analytic Surface," Am. 
M.S. Trans., XXIX, 471. 

5. SEGRE, B. "La cubique indicatrice de I'e'le'ment lineare projectif d'une 
surface," C.R., CLXXXIV, 729. 

6. GODEAUX. "Sur les lignes asymptotiques d'une surface et 1'espace r6g!6," 
Brux. Bull, ser. 5, XIII, 812. 

7. TERRACINI. "Sulla teoria delle congruenze W," Lomb. 1st. Rend. t LX, 
657. 



BIBLIOGRAPHY 311 

8. FUBINI. "Invariant! proiettivi, metric!, affini di una superficie," Boll 
d.U.M. It., VI, 113. 

9. LANE. "Contributions to the Theory of Conjugate Nets," Am. J., XLIX, 
565. 

10. LANE. "Power Series Expansions in the Neighborhood of a Point on a 
Surface/' Wash. Nat. Ac. Sc. Proc., XIII, 808. 

11. BOMPIANI. "Fascio di quadriche di Darboux e normale proiettiva in un 
punto di una superficie," Rom. Ace. L. Rend., ser. 6, VI 2 , 187. 

12. SEGRE, B. "Sur Integration d'un certain systeme d'6quations diffe*ren- 
tieUes," C.R., CLXXXIV, 268. 

1928. 1. FUBINI. "Una nuova generazione delle quadriche di Darboux," Rom. Ace. 

L. Rend., ser. 6, VII, 14. 

2. GROVE. "Transformations of Nets," Am. M.S. Trans., XXX, 483. 

3. SLOTNICK. "A Contribution to the Theory of Fundamental Transforma- 
tions of Surfaces," Am. M.S. Trans., XXX, 190. 

4. THOMSEN. "Sulle superficie minime proiettive," Ann. di mat., ser. 4, V, 
169. 

5. LANE. "The Projective Differential Geometry of Systems of Linear 
Homogeneous Differential Equations of the First Order," Am. M.S. 
Trans., XXX, 785. 

6. BEENKEN. Surfaces in Five-Dimensional Space. Chicago doctoral dis- 
sertation. 

7. CARPENTER. "A Theorem on Ruled Surfaces," Am. M.S. Bull., XXXIV, 
479. 

1929. 1. LANE. "On the Fundamental Transformation of Surfaces," Ann. of 

Math., XXX, 457. 

2. BOMPIANI. "Ein Analogon der Quadrik von Lie in der projektiven Fla- 
chen-Theorie," Math. Zschr., XXIX, 678. 

3. CAJORI. "Generalizations in Geometry as Seen in the History of Develop- 
able Surfaces," American Mathematical Monthly, XXXVI, 431. 

4. CECH. "Quelques remarques relatives & la g6ome*trie diffe*rentielle pro- 
jective des surfaces," C.R., CLXXXVIII, 1331. 

1930. 1. GRAUSTEIN. Introduction to Higher Geometry. New York: Macmillan 

Co. 

2. COOK. "Pairs of Rectilinear Congruences with Generators in One-to- 
One Correspondence," Am. M.S. Trans., XXXII, 31. 

3. MENDEL. "The Dimensionality of a Certain Linear Space in the Pro- 
jective Differential Geometry of a Variety in Hyperspace," Am. M.S. 
Bull, XXXVI, 219. 

1931. 1. SLOTNICK. "On the Projective Differential Geometry of Conjugate Nets," 

Am. J., LIII, 143. 



INDEX 

[The numbers refer to pages. Proper names are not listed, except when parts of titles.] 



Adjoint equation, 30, 159, 193, 211-12, 235 
Affine transformation, 245 
Ambient: definition of, 41 
of osculating plane, 124 
of space S(k, 0), 43 
8 k , 158 
Si, 182-83 

**+ 273 
Analytic curve: definition of, 3 

in ordinary space, 20 
parametric equations of, 3 
parametric vector equation of, 3 
in the plane, 13 

Analytic surface: definition of, 33, 220-21 
parametric vector equation of, 33 
proper, 33 
Analytic variety: definition of, 262 

parametric vector equation of, 262 
Anharmonic plane curve, 29 
Anti-ray net, 163 
tangents, 163 
Applicability, 36, 192, 221 
Associate axis congruence, 96 
Associate conjugate net, 95, 163 
Associate ray, 114, 163 
Associate ray congruence, 97 
Associated curve: of quasi-asymptotics, 62 

of a ruled surface, 61 
Associated linear complex, 302-3 
Asymptotic congruence, 51 
Asymptotic curves: definition of, 35 
belonging to linear complexes, 118, 205, 

208-9, 250 
curvilinear differential equation of, 36, 

214, 222, 245, 291, 298 
on a developable surface, 38 
flecnode surface, 59-60 
Laplace transformed surface, 140, 

143 

plane, 133 
quadric surface, 39 
ruled surface, 47 



surface in ,, 36, 66, 222, 241, 245 
surface in S n , 125-26 

projected into a plane net, 134 

straight lines, 36, 68, 108 
Asymptotic developable, 247 
Asymptotic osculating quadric, 79-80, 106, 

216 

Asymptotic regulus, 48 
Asymptotic tangent, 48-50, 74, 100 
Asymptotic tangent plane, 247 
Axial system of curves, 102 

see Union curves 
Axis: of a conjugate net, 95, 138, 214 

of a correspondence, 180, 190-91 

of the lines of curvature, 241 

of a special linear complex, 88 

see Cech, Linear complex: special 
Axis congruence, 95-96, 116, 139 

as congruence Ti, 95 
Axis curves, 117, 143, 146-47, 163 

indeterminate, 163 

plane, 163 

Axis-plane cone, 99-100 
Axis quadric cone, 115 
Axis tangents, 139, 146-47 

Bundle: of lines, 84, 150, 172, 189-90, 195, 

243-44, 252 

of nets on a surface, 118, 165 
of planes, 187, 191-92 
of quadrics, 21-22 

Canonical algebraic surface, 296 
Canonical cubic; see Wilczynski 
Canonical edges of Green, 91 ff., 116-17, 

163 
Canonical form for equations of: 

conjugate net, 139, 166 

pair of ruled surfaces, 279 

plane net, 162 

ruled surface, 44 ff. 

see Fubini, Halphen, Laguerre-Forsyth, 
Wilczynski 



313 



314 



PROJECTIVE DIFFERENTIAL GEOMETRY 



Canonical lines, 91, 93, 118 

pencils, 90-91, 202 

plane, 90 

point, 91 

tangents, 91, 93 

Canonical quadrie; see Wilczynski 
Cech: axis of, 90, 100, 116 

transformation of, 104-5, 116 
Central rays, 300 
Chasles: correlation of, 40, 283 
Christoffel symbols, 222, 237 
Circle, 229-30 
Class of a developable, 58 
Codazzi equations, 65, 222, 237, 241 
Coincidence curve, 15, 30 

point, 15, 29 

surface, 118 

Complete integrability, 68, 200 
Complex: definition of, 25 

of tangents to a surface, 197-98 

see Linear complex, Osculating linear 

complex 
Complex curve, 56 

points, 56, 61-62 
Cone: definition of, 259 

FJ of spaces 5(3, 1), 261 

see Axis plane, Cubic, Del Pezzo, Hy- 

pergeodesics, Segre, Seven-point 
Cone curves, 108, 163 
Congruence: definition of, 25 

belonging to a linear complex, 303-4 

ri,r,,82ff.,86 

in hyperspace, 127, 165-66, 181-82 

see Linear congruence, Osculating linear 

congruence, W congruence 
Congruentially associated net, 135-37 
Conjugacy of the second kind, 283 
Conjugate congruence, 168 

directions, 86, 265, 268 

invariant, 172-73, 214 

lines, 197 

points, 225 

tangents, 86 

triad, 213 

Conjugate net: definition in Sa, 86 
revised definition in S n , 121-22 

of cone curves, 163 

curvilinear differential equation of, 95, 
126, 164, 265 



on a developable surface, 162 

in the plane, 133 

of plane curves, 163 

of ruled surfaces, 217-18 

on a surface in S 4 , 126 
Conjugate relation : of net and congruence, 
150-51, 154, 158-59, 168, 211-12, 
234-35 

of surface and congruence Ti, 86 
"Consecutive": two meanings of, 270 
Contact of surfaces, 75, 115 
Covariant of a curve, 9 
Covariant quadric, 77, 79 

tetrahedron of reference, 24, 69 

triangle of reference, 16 
Cremona transformation, 104, 188 
Curvature: geodesic, 239 

mean, 223 

total, 223, 240-41, 246 
Curvilinear co-ordinates, 34 
Curvilinear differential equation of: 

anti-ray net, 163 

associate conjugate net, 95, 163 

asymptotic curves in S t , 36, 214, 222, 
245 

in conjugate parameters, 139-40, 
143, 298 

axis curves, 139, 143 

cone curves on a surface, 108 

congruentially associated net, 136 

conjugate net in asymptotic parame- 
ters, 95, 164 

conjugate net in S n , 126, 265 

developables of congruence p<r t 186 

Ti-curves, 84, 252 

recurves, 86, 253 

intersector curves, 278 

lines of curvature, 223 

net of curves, 34 

one-parameter family of curves, 34 

pangeodesics, 106 

parametric net, 34 

pencil of conjugate nets, 97 

plane curves on a surface, 107 

principal curves, 267 

ray curves, 141 

Curvilinear equation of a curve, 34 
Cusp, 10, 264, 266 



INDEX 



315 



Cusp-axis: of hypergeodesics, 101 
of a pencil of conjugate nets, 100, 117 
of the protective geodesies, 102 
of the Segre-Darboux pencil, 100 

Cusp-axis curves, 117 

Cusp-plane, 100, 189 

Cusp-tangents, 266 

Cuspidal edge, 37 

Darboux curves, 76, 100, 107, 114, 117-18, 
204-5, 216-17, 292 
in conjugate parameters, 164, 240 
directions, 76 
quadrics, 77, 85, 114, 119, 240, 253, 255, 

295-96 

tangents, 76, 98-99, 105, 109, 117 
Del Pezzo: cone of, 259, 261, 264, 281-83 
Demoulin: tetrahedron of, 109 ff., 119, 

203-4, 216 
Derivant net, 164 
Derivative point, 10 
ruled surface, 51, 60 
sphere, 230-31 
Derived net, 164 

Developable surface: definition of, 37 
of axis congruence, 129 
class of, 58 

condition for, 38, 40, 298 
of congruence Fi, 81-84 

T 2 , 85-86 

of flecnode congruence, 59 
history of, 286-87 
isotropic; see Isotropic 
parametric vector equation of, 37 
of ray congruence, 141 
special cases, 37 
total curvature of, 223 
of twisted cubic, 58 

Differential equations defining configura- 
tions : 

coincidence surface, 117 
congruence in S n , 150 

in S z , 297 

conjugate net in $a, 138 
in Green's form, 163, 298 
Laplace transform of, 149 
in plane coordinates, 165 
quadratic with equal invariants, 215 
cubic surface in 8 , 114 



curve in S n , 6 

in the plane, 5 
developable surface, 162 
flecnode surface, 53 
pah- of ruled surfaces, 279 
pair of surfaces hi 8 , 183 
plane net, 133 
quadratic net in 6 , 199 
quadric surface, 113 
ruled surface in *S 3 , 43 
in 8*, 60 
in S 6 , 61 

set of varieties in S n , 276, 280 
surface with property of Tzitz&ca, 251 

polar transform of, 254 
surface referred to an arbitrary net in 
8 t , 113, 187 
in affine theory, 245 
in metric theory, 222 
surface referred to its asymptotics, 67, 
113, 293 

in plane coordinates, 115 
second focal surface of a W congru- 
ence, 208 
surface in 4 , 282 

in S tt 267 
surfaces with intersecting tangent 

planes, 178 

transformation F in $ 8 , 172 
transversal surface of a congruence, 165, 

182, 285 

Differentiation formulas for: 
local line coordinates, 113 
local point coordinates, 111, 120 
transformation of parameters, 125 
Direction: definition of, 35 
Director curves : of a ruled surface, 40, 44, 

47,52 

of a variety V k+v 270 
of a variety V mt 272 
Director variety, 272 
Directrices; see Linear congruence, Wil- 

czynski 

Directrix congruences, 88 
Directrix curves, 112, 248 

indeterminate, 248 ff . 
Double point, 11, 14, 29, 73, 98, 105, 264 
Dual union curves, 102, 116 



316 



PROJECTIVE DIFFERENTIAL GEOMETRY 



Dualistic correspondence, 87, 115 
Dupin indicatrix, 36, 94 

Edge of Green; see Canonical edges 
Edge of regression: definition of, 37 
Eight-point: cones of a curve, 28 

cubics, 14 

nodal cubic, 14-15 

quadrics of a curve, 28 
Element of arc, 221 

see Protective linear element 
Element E r of a curve, 12, 41-42, 258 
Eleven-point cone of a curve, 30 
Envelope; see Axis-plane cone, Demoulin, 

Ray conic, Ribaucour, Segre cone 
Euler's equation, 102, 106 

F surface; see Surface F 
F transform; see Fundamental transforma- 
tion 

Five-point twisted cubics, 29 
Fixed point: conditions for, 4, 33, 108, 113, 

249 
Flecnode congruence, 51, 59-60 

curve, 52, 54, 61-62, 73-74, 114, 254 
points, 51-52, 59, 73-74 
coincident, 52 
indeterminate, 52 
sequence, 54 
surface, 52-53, 62, 205 
tangent, 52-53, 205 
transformation, 54 
Flex-ray: of hypergeodesics, 101 

of a pencil of conjugate nets, 98, 114 
Flex-ray congruence, 98 
Focal net of a congruence, 128, 150-51, 

154, 159, 198-99 

Focal planer of the axis, 146, 163 
of a line x u x v , 172 
of a line xy, 217 
of the ray, 141-42, 146 
Focal point: of the axis, 138, 142, 146 
of the directrix di, 249 
of a generator of a developable, 37 
of a generator S k of a V k+v 271-72 
of a line h, 186 
/!, 84, 252 
li, 86, 253 
p<r, 153, 215 
xy, 171, 175 . 



of the ray, 136-37, 141, 146 
of the tangent plane, 275 
Focal surface: of a congruence in S n , 128 

in ,, 296 
of a congruence ft, 81-84 

T 2 , 85-86 

of the flecnode congruence, 59 
of the ray congruence, 141 
Foci, focus; see Focal point 
Four-point conies, 30, 93 
Fubini: canonical form, 65, 69 ff., 113, 206, 

294 

definition of protective normal, 87, 102 
method of, 65, 289 ff . 
normal coordinates, 69, 242-43, 248, 

291 
Fundamental coefficients: first, 221 

second, 222 
Fundamental forms: first, 221 

second, 222 
Fundamental transformation : definition of, 

168 

analytic representation of, 169-71 
in S 8 , 172-73 

Ti-curves, 84, 86, 88, 107, 252 
r 2 -curves, 86, 88, 253 
ft, T 2 ; see Congruence 
Gauss equation, 65, 222 
Generator: of a developable, 37 

planar, of Q 4 , 195, 202 

rectilinear, of Q 4 , 195 

of a ruled surface, 32, 39 

S k of V k+1) 270-71 

*S(2, 1) of cone of Del Pezzo, 259 

8(3, 1) of the cone Fjj, 260 
Geodesic curvature, 239 
Geodesic curve, 107, 239 
Goursat; see Terminating 

Halphen canonical form, 28-29 
Halphen point: of a plane curve, 15, 28 

of a space curve, 22, 24 
Harmonic congruence of a transformation 

F, 168 

Harmonic conjugate net, 145-47, 163-64 
Harmonic invariant: of differential forms, 

265 
of a transformation F, 172-73, 214 



INDEX 



317 



Harmonic relation: of net and congruence, 
151-54, 156-58, 16&-69, 194, 235 

of surface and congruence T 8 , 87 
Harmonic triad, 213 
Homology, 190 
Hypergeodesics: definition of, 100 

cone enveloped by osculating planes, 
101 

cusp-axis of, 101 

flex-ray of, 101 
Hyperosculating conic, 14 

linear complex, 27 

Immersion in a linear space, 5 

in ordinary space, 16-17 

in the plane, 5, 13 
Indices of developability, 63 
Inflexion point: condition for, 11 

of a curve on a surface, 74, 81, 188 

of a nodal cubic, 15-16, 105 

of the ray-point cubic, 98, 101 
Inflexional tangents, 74 
Integrability conditions for: 

congruence in $ 8 , 297 

conjugate net in S s , 139 

pair of surfaces in 8 , 184 

plane net, 134 

set of varieties in S n , 281 

surface in 8 , 67, 69 
in affine theory, 245 
in metric theory, 222 

transformation F in $ 8 , 173 

transversal surface of a congruence, 182, 

285 
Integral curve, 6-8 

plane net, 134 

ruled surface, 43-44 

surface, 68, 70, 166, 182, 245, 251, 285 
Intersector curves, 278 

tangents, 278-80 
Intrinsic forms, 290 
Invariant, 9, 16, 27, 304 

see Conjugate, Harmonic, Laplace-Dar- 

boux, Weingarten 
Invariant integral, 102, 106 
Invariant lines of a homology, 190 
Inverse points, 225, 228, 251 
Involute curve, 56, 61 

points, 56 



Involution, 61, 176, 265 

Isothermally asymptotic surface, 118, 216, 

244,248 
Isothermally conjugate net, 26, 143, 145- 

47, 163, 210, 217 

Isothermally orthogonal net, 236-37 
Isothermic surface, 237, 252-53 
Isotropic developable, 222-23, 251 
line, 231, 251 

J net, 163 

Koenigs: conies of, 174r-75, 214 

theorem on plane nets, 134, 162, 254 
transformation of, 175 

Lagrange adjoint; see Adjoint 
Laguerre-Forsyth canonical form, 9, 28 
Laplace: case of; see Terminating 
Laplace-Darboux invariants, 124, 129, 

134, 162, 172, 200, 210 
equal, 124, 134, 136-37, 162-63, 175, 

210, 214, 237, 246 
see Tangential invariants 
Laplace equation, 122, 126, 128-29, 151, 
162, 169, 171, 193, 198-99, 208-9, 
223-24, 227, 233, 246, 251-52 
Laplace sequence, 129-33, 233 

see Periodic, Polar, Terminating 
Laplace transformation, 123, 127 if., 149, 
160-61, 171, 199-200, 209, 233, 235, 
299 

of a congruence, 299, 303-4 
Laplace transforms; see Laplace transfor- 
mation 

Levy transformation, 153-54, 194, 217 
Lie: polarity of, 104r-5 

quadric of, 78-79, 85, 105, 109 ff., 114, 

119, 203, 216, 254 

Linear complex: definition of, 25-26 
asymptotic curve belonging to, 118, 

205, 208-9, 250 
conjugate lines, 197 
curve belonging to, 27, 29 
first-order contact with a congruence, 

301 

null system of, 26, 208 
osculating; see Osculating linear com- 
plex 



318 



PROJECTIVE DIFFERENTIAL GEOMETRY 



Linear complex Continued 
second image of, 195 
special, 88, 194-95, 202, 300, 303 
W congruence belonging to, 208 

Linear congruence: definition of, 88 
classification, 196-7 
directrices of, 88, 196, 303 
in pencil of linear complexes, 88-89 
ruled surface belonging to, 62 

Lines of curvature, 223-24, 231 ff., 237 ff. 

Local tetrahedron, 24, 48-50, 55, 59, 71- 
73, 92, 110, 176, 187, 202, 213, 242, 
255 
characterized geometrically, 24, 94 

Local triangle, 13, 28, 174, 176, 258 

Local trihedron, 237 ff., 252, 255 

Minimal curves, 221, 231, 251 

surface, 223 
Moutard quadric, 115-17 

Net of curves: definition of, 34, 121 

Node; see Double point 

Normal of a surface, 221, 240, 243-44 

Null system: of a linear complex, 26, 208 
of osculating linear complex, 27, 55, 62 
of osculating twisted cubic, 23, 27 
of a twisted cubic, 19 

Order of: algebraic curve in 8 , 16 

developable surface, 58 

plane curve, 13 

surface in S 3 , 16 

variety in S n , 260-61 
Orthogonal net, 223 

spheres, 227, 229, 251 
Osculating circle, 231 
Osculating conic: of a curve in S^ 21 

of a plane curve, 14, 16, 28, 115 

of a twisted cubic, 19-20 
Osculating cubic of a plane curve, 16, 29 

twisted cubic of a curve in 8 , 20 
Osculating hyperplane of a curve, 12 
Osculating linear complex: 

of an asymptotic curve, 89, 118, 201-2 

of a curve, 26-27, 29 

of a ruled surface, 54-55, 201 

of a W congruence, 217, 301-2 
Osculating linear congruence, 205 



Osculating plane: definition of, 11 
of a curve in a Laplace sequence, 131 
curve on a ruled surface, 59 
curve on a surface, 80, 239, 259 
Darboux curve, 117 
Segre curve, 90, 117 
straight line, 36 
twisted cubic, 18-19 
Osculating quadric cone, 22 
Osculating quadric surface: 
of a curve, 28 

of a ruled surface, 48-50, 62 
Osculating regulus, 48, 50-51, 60, 203 
Osculating space: S k of a curve, 11, 41, 131 
$2*+! of a ruled surface, 43 
S(k, r)',see Space S(fc, r) 
Osculating sphere, 231 

Pangeodesics, 106ft 7 ., 117 
Parametric curves, 34 

net, 34 

tangents, 34 
Parametric vector equation of: 

a curve, 3 

developable surface, 37 

flecnode surface, 53 

ruled surface, 39 

surface, 33 

variety V 4 , 274 

*W 27 
V~> 262 

Pencil: definition of, 260 
Pencil of: conies, 174, 176 

conjugate nets, 97, 101 

linear complexes, 88-89, 303 

lines, 34-35, 190, 195, 258, 300; see 
Canonical pencil 

nets in a bundle, 118, 165 

plane cubic curves, 14, 104-5 

planes, 40, 61, 105, 115, 189, 192, 258 

quadrics, 117, 213-14 

spaces *S 8 , 260 

spaces S (3, 2), 260 

spheres, 228, 251 

classification of, 228 
Pentaspherical coordinates, 225-26 

coordinates of a sphere, 226 
Periodic Laplace sequence, 166, 254 
Permutability: theorem of, 168, 213 



INDEX 



319 



Planar system of curves, 103, 216 
Plane curves: condition for, 5 

forming a conjugate net on a surface, 
163 

on a surface in 8 , 107-8 
in S 6 , 283 

theory of, 12 ff . 
Plane nets, 133 ff., 162 

with equal invariants, 134, 136-37, 162 
Pliickerian line coordinates, 25-27, 88-90, 

112-13, 194 ff., 199, 203, 300 
Point equation; see Laplace equation 

invariants; see Laplace-Darboux in- 
variants 

Point-sphere, 224, 228, 251 
Polar planes, 197, 203, 216 
Polar sequences of Laplace, 159 ff., 201, 

233 

Polarity of Lie; see Lie 
Power of a point, 225 
Principal curves, 204, 216, 267, 283 

indeterminate, 268-69, 283-84 
Principal directions, 267-68 

tangents, 267 
Principal focal rays, 303 
Principal line of a curve, 31 

plane of a curve, 22, 31 

point of a curve, 31 

point of the tangent, 23 
Principal radii of curvature, 223 
Principal ruled surface, 60 
Principal spheres, 232-33 
Projective applicability, 191-92 

arc length, 102 

geodesies, 102, 118 

linear element, 192 

lines of curvature, 94, 107 

normal, 87, 92-94, 102, 202, 243, 253-54 
Projective differential geometry: definition 

of, 2 
Projective transformation, 1, 6, 44, 68, 

134, 173 

Projectively minimal surface, 119 
Projectivity between: bundles of lines, 189 

bundles of planes, 191-92 

lines and twisted cubic, 261 

pencils of tangents, 178, 180 
Projectivity on a line h, 179-81 

invariant of, 182 



Proper sphere: definition of, 224 
Proper surface: definition of, 33 
Pseudo-normal; see Projective normal 

Quadratic congruence, 192-94, 198, 210 

net, 192-94, 199, 210, 215, 232 
Quadrics: containing asymptotic tangents, 
103, 115 

with second-order contact, 75 ff., 240 
Quasi-asymptotic curves: definition of, 57 

associated curves of, 62 

equation of, 57, 60 

on a ruled surface in S t , 60 

on a surface in S 4 , 282 

R net, 146, 149, 164, 210, 217 

R relation; see Relation R 

R surface, 210 

Radical plane, 251 

Ray: of a conjugate net, 97, 163, 213 

of lines of curvature, 241 

of a plane net, 136 
Ray congruence, 97, 114, 141, 147, 163 

as congruence T2, 97 
Ray conic, 99, 114, 117 
Ray curves, 117, 141, 146-47, 163 

indeterminate, 163 
Ray tangents, 141, 146-47, 163 
Ray-point, 97, 108-9, 123, 127-28, 175 
Ray-point cubic, 98, 117-18 
Reciprocal congruences, 81 ff., 88, 97, 100, 
284-85 

lines, 253 

ruled surfaces, 284 
Regulus: asymptotic, 48 

common to three linear complexes, 197 

osculating, 48 
Relation R, 81, 118-19 
Ribaucour: transformation of, 193-94, 210- 

12, 235, 252 

Riccati equation, 8, 47, 58, 279 
Ruled plane, 195 

Ruled space, ordinary, 167, 194 ff. 
Ruled surface: definition of, 32 

parametric vector equation of, 39 

in space R$ t 197 
S 8 , 43 ff . 
S 4 , 60 



320 PROJECTIVE DIFFERENTIAL GEOMETRY 

Sannia: point of, 21, 25 L, M, N, 36 

Scalar, 3 C y , C g , 39 

Scroll directrix, 117 I y9 ^ 39 

Second image: of a linear complex, 195 ff. S [k, r), 41 

of a sphere, 227, 252 S (k, 0), 41 

Segre: cone of, 109, 120 R vff 51 

correspondence of, 1045, 116 A k n _ k , 57 

curve of the sixth order, 109 e, 68 
curves, 77, 100, 116-17, 216 -, Xf 69 

in con jugate 'parameters, 164, 240 <?, \p, 69 

directions, 77 I, m , 69 

tangents, 77, 99, 100, 117 . Q u , Q v , 79 

Segre-Darboux pencil, 100, 116 l lt I 2t 81 

Seminvariant, 8 FI, r, 82 

Sequence of Laplace; see Laplace sequence A, B, F, G, 84 

Serret: theorem of, 47-48, 58 d\, dj, 88 

Seven-point: cones of a curve, 21 ai, 02, 90 

quadrics of a curve, 21 e lf e*, 91 

Sextactic point, 14, 29 AT X , 95 

Singular space, 272 C x , C_ x , 95 

tangent space, 272 p x> 97 

Space Mi, 227 ff. p 1 1 1 

Space R*: definition of, 194 R^ t R v , 114 

dimensionality of, 194 N , 123 

Space S(k, r) : definition of, 41, 258, 262 x*> x_ l 123 

dimensionality for a ruled surface, ff, K, 124 

41-42 x r , x_ r , 128-29 

for a surface, 262 N , N_ , 129 

for a variety V M , 262-63 H ", K ^129 

Space S(k, 0), 41-42, 124, 258, 262 r> 139 r 

dimensionality for a ruled surface, 42 s v S_ v 140 

for a surface, 258 jp(u) jp(v) ^49 
for a variety V m , 262-63 ^^ ^ ' 

Space ^(2, 0), 42, 124-26, 201, 259, 265-66, ^ ^ 142 

268, 275, 285 r ,f > 150 

Space S(3, 0), 182-83, 261 ', S% 3), 164 

Space V(k, r), 262 h, 178 

Stationary osculating plane, 108, 114 X, Y, 193 

Stationary point; see Cusp ^3, 194 

ray-point, 108 (a, a>), 194 

Steiner's Roman surface, 120 Q 4 , 195 

Surface F, 265, 275, 285 , r, 198 

Symbols (in order of first occurrence) : A 8 , . . . , A^ 202 

*S n , 1 12, 203 

* /pi 3 Rif R%> 223 

C x , 6 K, K mt 223 

? r , 12 M 8 , 227 

2, 12 &,, 227 

C^, C v , 34 plt ^ 239 



INDEX 



321 



Symbols (in order of first occurence) 
Continued 
Vl 256 
V (k, r), 262 
0, Ai, Ni, T, 293 

Tacnode, 266-67 

Tangent line:, definition of, 10 

of a curve on a surface, 34-35, 258 
in metric theory, 238 

of the flecnode curve, 52 
Tangent plane: definition of, 35 

of a developable surface, 38 

equation in &, 35, 221 

of a net, 150 

of a ruled surface, 40 

as space S(l, 0), 41-42, 258 
Tangent space S m of V m , 262 

S k+ i*V k+v 272-73 



Tangent space S* of a ruled surface, 43 
s zk+i of v k+v 271-72 

(*+!) (m+l)-l f V k+m> 273 

Tangential equation of a net, 142 

invariants, 142, 146-^7, 149, 163, 166 
Terminating Laplace sequence: 

case of Goursat, 130-32, 205, 252 

case of Laplace, 129-33, 205 

mixed case, 130-31 

Transformation; see Affine, Cech, Flec- 
node, Fundamental, Koenigs, La- 
place, Levy, Projective, Ribaucour 
Transformation connecting: 

asymptotic and conjugate parameters, 
143-45, 164 

cartesian and Fubini's normal coordi- 
nates, 241-43, 248 

general and local trihedrons, 238 

lines of R* and points on Q 4 , 194 ff . 

local tetrahedron and trihedron, 255 

planes of two bundles, 188 

points of Af 8 and points on Q 8 , 227 ff . 

points of two surfaces, 177 
in different spaces &, 186 ff. 
in space 3, 183 ff . 



two tetrahedrons, 50, 92, 110, 176 
Wilczynski's and Fubini's canonical 

forms, 113, 293-94 
Transformation preserving: 

equations of a congruence in 89, 297 

conjugate net in , 145 
Fubini's canonical form, 70 
Transformation of proportionality factor, 
7, 68, 123, 145, 150, 157, 160, 242-43, 
252, 290 
Transformation of variables: 

dependent, for a ruled surface, 44 ff . 
for a set of varieties in S n , 276-77 
independent, 8, 46, 123, 125, 145, 180, 

276, 281, 290 
Transversal surface of a congruence, 165- 

66, 182 

Triple point, 76, 264, 267 
Twisted cubic in S z : definition of, 17 
determination by six points, 18 
five-point, 29 
null system of, 19 
osculating, 20 

parametric equations of, 18 
in nded-surface theory, 62 
in surface theory in S n , 261 
Tzitz4ica: property of, 247-18, 250-51, 
254 

Union curves, 102, 107, 116, 190-91, 241 

Vector, 3 

Veronese: surface of, 269, 283-84 

W congruence, 60, 140, 146, 299 
belonging to a linear complex, 208 
Fubini's theory, 205 ff . 
with ruled focal surfaces, 208-9, 216 
in space Rt, 209 ff. 

Weingarten invariants W (u \ W (v \ 140, 
142, 149, 253 
TF,298 

Wilczynski: canonical cubic of, 295 
canonical form, 65, 113, 293 
canonical quadric of, 80, 119, 295 
directrices of, 88, 90, 112, 114, 202, 204 
methods of, 9, 64, 288, 292 



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